Why do my fried eggs start browning very fast?
Your pan is too hot. To perfectly fry an egg--without browning--you want medium to medium-low heat; what you want is the heat at which butter will juuuust sizzle. And don't use oil.
The easiest way to ensure that the residual white is cooked on top (assuming you want sunny side up) is to put a lid over the egg when it's almost cooked. Optionally splash a tablespoon of water into the pan (but not on the egg) immediately before putting the lid on.