I pan fried chicken cutlets and stuck the thermometer in the thickest part and it read 165 degrees but when I ate a bit there was a small spot of pink, slightly raw tasting chicken, kind of on the side. These were fairly thin pieces. Does the thermometer average the temperature and while some parts may be quite hot can it still have colder spots? How do you ensure that all of the cutlet is cooked if the thermometer is missing the cold spots? My thermometer seems reliable when I use it on roast or baked chicken.
If the raw spot was in a thin area, I'd suggest that it was caused by the pan frying method.
A thin spot could leave a gap between the surface of the pan and the meat itself, and so not absorb much heat.
To reduce these indentations, you can mechanically reshape the meat before cooking (e.g. squash down the higher parts).
In addition, or instead, as soon as you start cooking, it's a good idea to use the spatula to briefly press the meat onto the pan's surface. This will flatten the bottom face of the meat and give it uniform contact with the heat.
By the time you turn it over, the meat will have become firm, so you won't be able to fully flatten the second face against the pan, but that's okay as the thin parts will already be mostly cooked and the parts that still need heat will be in contact with the cooking surface.
The only way to accurately assess done-ness is with a thermometer (though it is difficult to measure the temperature accurately in thin pieces of meat). Chicken can be pink, and also be cooked. Visual signs and flavor perception are much less accurate. You say your thermometer "seems" accurate, but have you calibrated it? This should be your first step. Then, making sure you are using it correctly should be next. Finally, if you are using proper cooking technique, it is safe to assume that when the thickest part of the product is cooked, the rest is cooked.
(one of) The problem when cooking a chicken breast is that it does not have a consistent shape, one part is thicker than the other.
This will often result in either undercooking (thickest part) or overcooking the meat (thinnest part).
If you put the thermometer in the thinnest part of the meat, then it will register a good temperature, while the other part will not be there yet.
Couple of suggestions that will help you: first, don't put a too cold meat in the pan; get it out of the fridge, like 30 minutes before so that it get to an even temperature; second, you can pound the breast so that it has a consistent thickness, so that all part will cook at the same time.
It is absolutely normal that the chicken is not the same degree of doneness throughout. The people who write guidelines know that and have accounted for that in their guidelines.
It is not that there is some magic temperature at which meat will change from "will always make you sick" to "will never make you sick" and they give you that temperature. Rather, they give you the answer to the question
"If I am a home cook and stick a thermometer into the roughly thickest part of my meat, what temperature should the thermometer show so I can count the meat as safely cooked?"
They are aware that not all people's thermometers are perfectly calibrated. They are aware that you won't always stick the thermometer into the perfect spot. They are aware that some parts of the meat will cook at a different speed from other ones. And so on. But they have made a complicated model that takes all this into account when telling you what temperature to look for on the thermometer display.
They also had a point in making it about temperature. Even if it looks more pink than usual, or has colored juices, or whatever, that doesn't make it unsafe. These factors are correlated with doneness, but are not a precise indicator. So just follow the guidelines as written. If your thermometer showed the required temperature, you are safe.