Cut each slice of lemon into 6 wedges. Sprinkle with the sugar and set aside.
Lay out 4 slices of bread. Top each with 2 thin slices of mozzarella, then a few wedges of lemon, then one more piece of mozzarella. Top with the remaining slices of bread. Spread half of the butter on the top slices of bread and sprinkle each sandwich with 1 tablespoon of Parmesan cheese.
Heat a large skillet over medium heat. Place the sandwiches Parmesan-side down in the skillet. Spread the remaining butter on the bare slices of bread and sprinkle each one with 1 tablespoon of Parmesan cheese. Cover the skillet and cook until the bread and Parmesan are golden brown, about 2 minutes. Flip the sandwiches and cook until golden brown and the cheese inside is melted. Serve warm.
If using no-salt or low-salt mozzarella cheese, the filling may need to be seasoned with kosher salt.