In a small bowl, combine 2 tablespoons softened butter, 1 tablespoon finely grated Parmesan cheese, and 1/2 garlic clove grated (or 1/4 teaspoon garlic powder). Stir until smooth.

Spread the butter mixture evenly on one side of each of the 4 slices of sourdough bread.

Place the 2 ounces sliced mozzarella and 2 ounces sliced white cheddar on the unbuttered side of two slices of bread, dividing the cheese evenly between them.

Top each cheese-covered slice with the remaining bread slice, with the buttered side facing outward.

Heat a griddle or large skillet over medium-low heat until hot but not smoking.

Carefully place the assembled sandwiches onto the hot griddle or skillet and cook until the bottom is golden brown, about 4–5 minutes.

Flip the sandwiches using a spatula, press lightly with the spatula if desired, and continue cooking until the second side is golden brown and the cheese is completely melted, about 3–5 minutes more. If needed, cover the skillet briefly to help the cheese melt.

Remove the sandwiches from the pan, let rest 1 minute, slice, and serve immediately.