One of the methods I use to cook chicken in general and especially chicken breast is as follows:
put the chicken into a frying pan. Pour 1/2 a cup of water and let it cook on medium heat for as long as two or three tablespoons of water remains. I use small portions of onion to get rid of its bad smell.
The problem with this method is always get a dry meat. What method gives me a moist and juicy meat so that it melts in the mouth?
The reason why your meat is dry has less to do with whether you cook in liquid and more to do with the temperature you cook the meat to. The proteins in meat will squeeze out the water in them as you increase the temperature you cook them at. If you are cooking your breasts in boiling water then that can result in meat that is cooked far above well done. Here is a link that relates moisture loss in meat vs cooking temp:
http://www.seriouseats.com/2010/03/how-to-sous-vide-steak.html
This link is for beef but the basic principle applies to chicken as well.
Water is able to transfer heat much more efficiently then air given a similar temperature. This means that your chicken is ironically getting overcooked because of the water you added. A standard technique people use to cook small cuts of meat is to brown both sides at a high temperature and then lower the temp to cook the inside without over cooking the outside. Next time just try to cook without water and just lower the temp instead. Flip the meat often to ensure even cooking.
If you want truly "melt in your mouth" chicken you will need something with a bit more fat. Chicken breast is very lean protein and at best can be "juicy" but that is it. Chicken legs and thighs however are higher in fat content and have collagen which can result in very tender chicken. The legs and thighs can also be cooked to a much higher temperature without drying out as much as chicken breast. Consider cooking dark meat for tender chicken.
Chicken breast is very unforgiving of being overcooked, which is what is going wrong. Breast meat is very lean, cook it too long and all the moisture goes. The method you are using will not give you good, consistent results - how long it will take water to evaporate depends on too many factors, and it also doesn't control for the size of the breasts which can very significantly. What you need to do is to measure the temperature of thickest part of the breast as it is cooking and remove it when it reaches 170F/77C.
Something that might help is to salt the chicken and leave it in the refridgerator for a little while before cooking it. Serious Eats has a good article about brining and salting.
Here's why it works:
Muscles are made up of long, bundled fibers, each one housed in a tough protein sheath. As the turkey heats, the proteins that make up this sheath will contract. Just like a squeezing a tube of toothpaste, this causes juices to be forced out of the bird. Heat them to much above 150°F or so, and you end up with dry, stringy meat.
Salt helps mitigate this shrinkage by dissolving some of the muscle proteins (mainly myosin). The muscle fibers loosen up, allowing them to absorb more moisture, and more importantly, they don't contract as much when they cook, making sure that more of that moisture stays in-place as the turkey cooks.
The author recommends salting instead of brining though, since it keeps the flavor better:
Use kosher salt. Salt your meat liberally (it should look like a light snowfall on the bird). Place the bird on a plate in the fridge overnight and loosely cover it with plastic or cheesecloth. Rinse if desired to remove excess surface salt (I skip this step because I like salty skin). Pat dry. Roast as desired. For even better results, carefully separate the skin from the breast and thighs and rub the salt directly on the meat, under the skin.
The article is about turkey, but the author uses chicken as an example. Since chicken breasts are much smaller, you probably don't need to leave the salt on as long. This article suggests that around half an hour might be long enough. You'll probably want to experiment a little bit.
Chicken breasts by themselves are pretty hard to keep moist. Your best bet is cooking with a thermometer to avoid overcooking.
However, my best chicken breasts are always when cooking a whole chicken. The trick is to rest the roast upside down, and let it rest like this for at least twenty minutes. Or even cooking it upside down if you are impatient will give good results flavor-wise (but the roast won't look as nice).
This lets the juices flow down to the breasts, giving you amazingly moist white meat.
Trick learned from Maggie Beers.
Cook it at slow heat for more than an hour. Cover the utensil tightly to hold the steam. Overnight marination can help in getting uniform taste and make chicken quite tender.
You mentioned frying, but one method I like for baking chicken breast is to do so at higher heat, for a shorter amount of time. 20-22 minutes at 450 degrees Fahrenheit works well in my experience (making small quantities, marinating for 15-90 minutes beforehand).
I eat the chicken breasts from Costco (the Kirkland ones that are individually wrapped, in a salt solution), so here's my advice for those or any similar chicken breasts:
Frying:
Cut the chicken breast up into multiple segments. You can cut it once lengthwise, or 6-8 times crosswise. I usually do the latter because it's easy to mess up the lengthwise cut and end up with one thick piece and one thin piece. If you just try to fry the entire thing, it takes too long for the center to be fully cooked, and most of it ends up dry. Cutting it greatly reduces the cooktime and also improves the quality of your meal.
Baking:
Heat oven to 400 F / 204 C. Wrap the breast in aluminum foil, with seasonings and anything else you want to bake alongside it (e.g. peppers, onions, lentils). Bake for 25-30 minutes if it's thawed, 40-45 minutes if frozen (please make sure to measure the internal temperature yourself before eating!). You should adjust times based on how you prefer your chicken. If you don't want to keep track of the preheating you can put it in and wait 50 minutes (once again, measure internal temp, I don't want you to get sick).
I usually just bake it frozen, because it's so easy to take a frozen pre-wrapped breast out of the freezer, open it, wrap it, and toss it an into the unheated oven. You also don't have to deal with the liquid in the wrapper if it's frozen.
You are way overcooking your meat. I know some people want to overcook chicken meat for safety reasons, but here's two ways I cook chicken breast for very juicy meat:
Step 1: right when you buy your chicken breast, before putting them into your refrigerator, salt them! Salting them way before will make them much juicier later.
Step 2: a while before cooking, take the meat out of the refrigerator to adjust to room temperature.
Step 3: This depends on whether you want to cook it in fat, or in water.
Method 1: Cut the chicken breast however you want (or don't cut it), and cook it on good heat but not so hot that your butter burns. From time to time check the inside of the meat to see the color. It will cook from the outside in. Stop cooking before the meat is bright white. Now let the meat rest for a few minutes, and check the inside of the meat again to see if it's cooked. Don't expect it to look stark white and dry inside, because guess what, then it will also taste white and dry. It should have a soft white color, and be moist inside. The cooking time will depend on how big your pieces are. If you cut the breast into about 10 pieces, then this may take only 3 minutes to cook. If you've used the breast whole, it will take longer and you'll have to use lower heat not to burn the butter. After you've cooked them in this way for a few times, you get a feeling for it. Serve the remaining butter as a sauce, and make sure to get all the browned bits in the pan, they have the best taste. Or use the remaining butter as a basis for a sauce.
Method 2: If you want to cook chicken by boiling it, don't boil it in boiling water for several minutes. It will taste totally dry. For example for a thai chicken soup, I bring the soup to a boil, then turn off the heat, and then add the uncooked chicken pieces. The remaining heat of the soup is more than enough to cook the meat well if the pieces are thin, but it will not overcook (always check for doneness before serving of course).
The salting and bringing the meat up to room temperature helps, but if you overcook the meat later, it's all for nothing. The key is to not be afraid to undercook it. Better undercook than overcook, because if you undercook you can always cook it a little extra, but if you overcook it's impossible to undo it. If you've never tasted perfectly cooked chicken breast, then perfectly cooked chicken breast is going to look and feel undercooked to you, but as long as the meat inside does not look raw, you're good.
You can also consider switching to chicken thighs. You can cook those way longer and they'll still taste good. Though I would not recommend cooking them in water, because they'll be tasteless. By far the most tasteful basic way to cook chicken is just with salt and butter.
Simply cover your chicken breasts with streaky bacon. The salt in the bacon and also the fat from the bacon will keep the meat juicy.
