Heck yes - baste those eggs! I do the same thing with scallops; really good. The temperature of that butter or other fat/oil is going to be a lot hotter than what’s required to set the yolks, so it shouldn’t really take all that much…
I do think eggs are very satisfying. Most days now I just have black coffee in the morning and then at some point a few hard-boiled eggs or fried eggs. And then one ‘big’ meal in the evening.
For me, fat is very satisfying, as is protein. I’ve seen quite a few people say that just fat really doesn’t do it for them - and eggs have a nice mix of protein and fat, so eggs should pretty well cover things…