In a small bowl, stir together 2 tablespoons softened butter, 1 tablespoon finely grated Parmesan, and 1/2 garlic clove grated (or 1/4 teaspoon garlic powder) until smooth.
Spread the butter mixture evenly on one side of each of the 4 slices of sourdough bread.
On the uncoated side of two slices, layer the 2 ounces sliced mozzarella and the 2 ounces sliced white cheddar (use all the cheese, dividing it between the two sandwiches).
Top each cheese-layered slice with the remaining buttered bread slices so the buttered sides face outward; press gently to seal.
Preheat a griddle or large skillet over medium-low heat until hot.
Place the assembled sandwiches in the skillet, buttered side down, and cook until the bottom is golden brown, about 5 minutes.
Carefully flip the sandwiches and continue cooking until the second side is golden brown and the cheese is completely melted, about 5 minutes more.
Remove from the skillet, slice as desired, and serve immediately while hot.