In a large bowl, whisk together the eggs, milk, salt, pepper and hot sauce until evenly combined.
Melt 2 tablespoons butter in a large non-stick skillet over medium-high heat. Once melted, add the whisked eggs and cook, stirring often, until their a soft scrambled consistency, but not runny. Remove from the heat and let cool slightly.
To assemble the grilled cheese, lay out the bread slices in a single layer. Place a slice of cheese on each slice of bread (top and bottom). Divided the scrambled eggs among 4 slices of bread (the bottoms). Top with either bacon or sausage or both and then sandwich together with the other half of the bread (tops). Spread the outside of the bread with softened butter.
Heat a griddle or skillet over medium-high heat. Once hot, place the grilled cheese, buttered side down, and spread the opposite side with butter. Cook until golden brown and crispy on the first side, about 2 minutes. Flip over and continue to cook until the cheese is melted through and the second side has browned, about 2 minutes. Repeat with the remaining sandwiches and serve with ketchup or hot sauce on the side. Enjoy!