Oeufs au Miroir : this is just the best way to make a fried egg with a soft yolk. It's far easier to control and hit the perfect doneness level than with the traditional technique, and the white is more set. (with a normal fried egg you have the dilemma that you want the yolk just barely cooked, but undercooked white is disgusting, so you have to flip the egg just briefly and try not to break the yolk and be careful not to overcook it).
The technique is very simple : heat a nonstick pan to medium high as you would for a fried egg. Put butter in pan and let it get very hot but not browning. Gently crack in eggs. Immediately put a lid on the pan and turn it down to medium - medium low. The carryover heat in the pan should brown the bottom of the eggs well, you should hear it crackling. Do not take off the lid if at all possible - you need the steam in the enclosure to stay trapped inside. You want to judge doneness with your ears. When the crackling slows and the eggs become quiet, they're probably done. The top of the egg should be glassy and smooth (hence the "miroir") the yolk should be just starting to gel like a creme anglaise. If you like you can flip the egg over when serving so it looks like a fried egg (one side is fried, one side is steamed).
The traditional French technique for Oeufs au Miroir calls for adding a few drops of water to the pan when you crack them in, to aid in the steaming. I find that our shitty over-large eggs are so full of water that this isn't necessary and don't usually bother.
Scrambled eggs : most Americans make disgusting overcooked rubbery scrambled eggs; the traditional French technique for scrambled eggs calls for cooking them very low and slow with added dairy, which makes them almost a soup, which I also find disgusting. I think the sweet spot is somewhere in the middle, I want them cooked through, no raw or runny bits, but just barely set.
The hard thing about cooking eggs is that they cook very fast, so they quickly go from "just right" to "overcooked". This is why so many professionals are terrified of cooking eggs. Part of the difficulty is that if you plate them hot, they will carryover so even if they were perfect in the pan, they are overcooked when you eat them.
I actually think both of these things can be addressed pretty easily with the application of reason and logic. The way you solve the difficulty of eggs flying so fast past the target doneness is just to cook them slower. Similarly the way you solve the difficulty of carryover is just to plate them at closer to room temperature. Hence we have a very slight modification to traditional scrambled egg technique that I think makes it much easier to do well :
Heat a pan to medium high and butter, just like for a fried egg. Crack eggs into pan, they will be crackling and cooking very fast at this point. Immediately turn the pan *OFF*. We are going to scramble them entirely with the residual heat in the pan and let it cool down. Stir the eggs a bit gently, the pan should still be very hot at this point and cooking fast, but also cooling fast. You can control the amount of cooking a bit by the amount of stirring - more stirring = less cooking. The eggs will be done in about a minute; as they get close to done the pan should be coming down to lukewarm, so you don't have so much risk of flying past doneness. Plate and again stir vigorously so they don't trap too much heat as they sit on the plate.
Breakfast scrambled eggs I like just plain. For dinner scrambled eggs, add a tiny dash of cream and a tablespoon of grated parm part way through cooking. Parmesan is an operator which converts eggs into dinner. Top with chives or parsley.