I grew up with coddled eggs (the little aluminum dishes that fit in a container with boiling water underneath), with soft-boiled eggs (in their cute little egg cups and the top chipped off) - though as I got older it was mostly scrambled (dry, not runny) or fried (over easy, yokes still soft, preferably a bit runny, whites completely cooked).

What is it about poached eggs that makes this a desirable way to cook them? Is it what you eat them with or how they're served?