Yep, make them basted and order them basted all the time.
I like the yolks runny, but not the whites, and this is the best way to do it without the danger of flipping them and breaking the yolk.
Basted, baby, basted!
I asked for basted on Sunday at semi fancy brunch place. The waiter said “sure, uh, wait a minute, we can’t do that. How about poached?” It was really busy and my daughter (actually most of them) were clambering for attention so I didn’t have the opportunity to drill down.