When I want a thick, round egg for a sandwich, be it runny, or hard yolk, I just use my poaching pan. I can make from 1 to 4 eggs at a time, and they come out perfect. Sometimes, I even make deviled eggs from the hard-poached eggs. It makes a unique presentation. And from the butter added to the poaching divets, the eggs are buttery good in flavor too.
I have egg rings that are round, and two that are heart shaped.. But DW won't eat a fried egg unless it's cooked medium over. Egg rings are hard to use if you have to flip the egg, though I can do it if I'm very careful, as the handles swivel. It's a tricky thing to do though.
Seeeeeeya; Chief Longwind of the North