There are two different techniques possible here – obviously, simply frying for a longer time will cook the yolk more thouroghly. You might order this as “sunny-side up, well done”.
However, what you achieve by cooking with a lid isn't really considered just frying, but Basting, so what you want to ask for is probably a basted egg.
In most style of omelette vegetables are optional. The obvious exception is the Spanish tortilla, where sliced or grated potato adds strength, but that's cooked more slowly and not turned.
Tomatoes in particular are tricky, as they're so wet. The same would apply to courgette, for example. Large quantities of veg also make it harder, as the egg will be too ...
You take raw eggs and crack them into the stock in the beginning, right after adding the COLD water. Put everything into the pot, add cold water (using cold water allows all of the plants to release all of their flavonoids), and then you crack the eggs into the pot, breaking up the shells in your hands and throwing them in as well, and then stir it ...
It's pretty safe.
Just boil each element the amount of time it requires, and if the eggs go in after the vegetables are already in, make sure that the eggs are at least room-temperature in order to prevent the water from stop boiling.
It's actually a thing to hard boil eggs in a hot pot, which as you know, is usually enjoyed with a variety of different ...
My mother used to put the eggs in a plastic bag in with the potatoes.
If you want hard boiled eggs, as she did, you can put them in with the cold patatoes and cold water. Take them out 10 minutes after the pot comes to the boil.
The plastic bag in case the eggs cracked. And because our eggs can be dirty, it also keeps the eggs from getting the veggies on ...
As most ways** of cooking eggs call for cooking by time, I wouldn't recommend it.
There's nothing, however, to stop you from fishing out the eggs after they're done, and then warming the water back up to cook your vegetables.
This is easier than than trying to remove one item while leaving the other to continue cooking, or experimenting with correct timing ...
Yes, you can do this. Be aware that:
if you add lots of cool items to boiling water it will take longer to start boiling again, which may impact cooking time a little.
I can imagine vegetables with very strong colour (beetroot) or flavour (garlic?) might diffuse through the eggshell and give you a pink or slightly garlicky egg.