I don’t know if I have a true favorite egg dish, but my most common way to eat them as a breakfast is to cook 4 XL or jumbo eggs sunny side up or very gently over easy (maximum wet yolk please), tear up 4-5 strips of freshly cooked bacon, and mash the bacon into the eggs on the plate, then scoop the yolky-white-bacon mass onto toast which has been too heavily (in the opinion of some curmudgeons) buttered.
Then cut bites out with a fork and enjoy.
But I also like them pretty much most ways for breakfast, whether scrambled, hard fried, omelet (sautéed mushrooms, onions, thin-sliced ham and cheeses, pleases) etc.
For non-breakfasty stuff, I do love deviled eggs and my current favorite version of that is a sort of Mediterranean style (more hummus than mayo, chopped olives, feta crumbles) that I learned of here when folks were showcasing nice recipes for deviled eggs.
ETA that deviled egg thread is here.