Pictured above is the Lodge 8 inch griddle. It's meant to be a serving griddle; The wood underliner is sold separately.
That griddle is my designated "egg pan." It's best to pick one pan to use for eggs, and stick with it.
The size is perfect because it doesn't "hang over" the small burner, and eggs have a bad habit of sliding to the side of the pan.
Season it about 3 or 4 times before you start cooking eggs, and then the first several times, they will probably still break.
After you cook the eggs, put some bacon grease or butter on the empty pan, into the oven at 250. This will "bake in" and strengthen the seasoning.
Preheat the pan on the stove, then melt some butter, then add eggs. Use a teaspoon to baste the top of the egg with the pan juices.
Cook until they turn white. Use a thin, METAL spatula, tilt the griddle, and ROLL over the egg, rather than pick up and drop.
Even with the above steps, your yolk still might break. But the essentials are preheating, butter, and a well seasoned griddle.