I have fried egg on toast quite a bit for breakfast and use a sandwich maker to do it. I marg/butter the toast on both sides before I put it on the (cold) flat surface then turn it on. When the first side is cooked enough, I turn it over, put a slice of cheese on the cooked side and butter (marg) the section of the pan where the egg is going to go. (not much butter/marg needed as the sandwich maker is hot and the fat melts very quickly.) I crack the egg slowly so the white only comes out and then I let the yolk fall into the setting white. By the time I have cleared away the rubbish and put away the marg/butter, the egg white is normally done sufficiently to turn it off. Sometimes if the white is too thick, I have to move the albumen off the cooked white to speed things up a bit. Still very soft and the yolk beautifully runny by the time I scoop the cheese toast out on to the plate, topped with my lovely egg!! A good dob of tomato sauce or tomato jam finishes the plate off nicely.
An older/stale egg means that the albumen runs further and so browns more and goes all crunchy instead of nice and soft whites. Also, I bring the egg to room temperature before cooking.