I often cook eggs in a 1-qt pan with a glass lid. I make sure the pan is hot enough that the egg sizzles continuously when I drop it in, and then leave the lid off just long enough to season the egg. Then I put on the lid, and wait for the white above the yolk to turn translucent (which I can see through the glass). At that point, the yolk is a little runny; shortly afterwards when the white it opaque, the yolk will be congealed on the outside but still a little runny on the inside, and not long after that, the yolk will be solid. Depending on what I'm in the mood for, I take it out at a different time.
The bottom of the egg is always browned, and the white is fluffy and full of holes like bread because of all the steam trapped inside. I'm not sure why your egg isn't turning out the way you like. It sounds like it might not have cooked fast enough on top, but the bottom got overcooked and over-hardened and dried out.
I'd suggest trying a few things:
- Don't check on the egg. Opening it up to check on it will let out all the steam. You might get a few wasted eggs at first while you get the hang of the timing, but it's not going to turn out right if you keep opening it up.
- Heat up the pan more before you start. You can always turn it down while it steams, but I find that cooking it when the butter just starts to sizzle isn't hot enough; I make sure it's hot enough to make the egg sizzle continuously once it hits the pan.
- Use a smaller pan. A big pan will take a while to fill with steam and heat all the way up.