Working in 2 batches, melt 1 tablespoon of butter in a large skillet over medium-high heat. Crack 3 eggs into the skillet and season with salt and pepper, to taste. Cook until the whites are nearly set, about 3 minutes. Carefully flip the eggs and continue cooking until the whites are completely set, about 1 minute more. You're going to top each portion of the casserole with a fried egg. Repeat with the remaining 3 eggs and 1 tablespoon butter. Top with Gruyere/Emmentaler Cheese Mixture and it in the oven for a few minutes until cheese melts.