What would you call this way of cooking an egg?
This would generally be known as a variant of &quot;basted eggs&quot;.
The idea of basted eggs is to apply heat from the bottom (from the pan) without flipping, but additionally cook the tops with liquid. The two ways of doing this are (1) to spoon hot fat over the eggs, or (2) to add water and cover the pan to allow steam to cook the tops.
My normal method of frying eggs is similar to this one, but at lower heat and cooking for longer uncovered. If you get it right, the white ends up fully opaque while the yolk is still deep yellow.