Mix 2 tbsp mayonnaise with 2 tbsp grated parmesan and set aside.
Line up sourdough bread slices in pairs to fill.
Arrange 3 slices of gouda on top of one slice of each sourdough pair.
Top with the bacon, breaking it into pieces that fit the bread size.
Top the bacon with 3 slices of sharp cheddar and then sprinkle the remaining parmesan on top of the sharp cheddar.
Close each of the three sandwiches with its sourdough pair.
Spread the mayonnaise and parmesan mix on the outside sides of each of the assembled sandwiches.
Heat a large skillet to medium and place your sandwiches in the skillet once hot. Top the skillet with a lid or a piece of foil and let sandwiches sit undisturbed 3-4 minutes.
Flip the sandwiches and cook for another 3-4 minutes, until bread is golden and crusty and the cheese has melted.
Remove sandwiches and let them rest on a cutting board or plate while you cook the egg.
Frying the eggs
Put 2 tbsp of oil in the hot skillet. Crack an egg in the oil and tilt the pan to one side while the egg cooks. Using a spoon, scoop up the hot oil and drizzle it on the top of the egg to cook the top of the egg white while the bottom becomes crispy and lightly browned. Cook the egg like this for 3-4 minutes, until you reach your preferred done-ness and then place the egg on top of one of the grilled cheeses. Repeat steps to cook the remaining eggs and top the remaining sandwiches.
If you don't want to cook bacon or don't have any on hand, thin sliced prosciutto or thin sliced honey ham are great substitutes.