Preheat the oven to 350°F. Line a baking sheet with parchment paper.
Spread 1 tablespoon softened butter on a piece of bread. Spread 1 ounce goat cheese on the other side. Heat a cast-iron skillet over medium-high heat. Place the bread, butter side down, in the pan.
While the bread toasts, sprinkle 2 ounces sharp cheddar on top of the goat cheese and then top with another piece of bread. Place 2 ounces fresh mozzarella on top of the second slice of bread.
Spread 1 tablespoon butter on a third slice of bread, then place it, butter side up, on top of the mozzarella. When the first piece of bread is golden brown, flip the sandwich over and toast until the other side is golden brown.
Transfer the sandwich to the prepared baking sheet. Repeat the process with the remaining bread, cheeses and butter. When both sandwiches are assembled on the baking sheet, transfer the sandwiches to the oven and bake until heated through, 5 minutes. Cool slightly before eating.