01 - Melt 1 tablespoon unsalted butter in a small skillet over medium heat. Add the thinly sliced apple, light brown sugar, ground cinnamon, ground nutmeg, and a pinch of salt. Cook, stirring occasionally, for 3-4 minutes until the apples are tender but retain their shape. Remove from heat and set aside.
02 - Arrange the bread slices. Evenly spread the softened unsalted butter on one side of each slice. Place two slices, buttered-side down, on a clean surface. Layer half of the sharp cheddar cheese and half of the mozzarella cheese onto each of these slices. Distribute the prepared spiced apples evenly over the cheese. Top with the remaining bread slices, buttered-side up.
03 - Preheat a nonstick skillet or griddle over medium-low heat. Carefully place the assembled sandwiches onto the hot surface. Grill for 3-4 minutes per side, applying gentle pressure with a spatula, until the bread achieves a golden-brown hue and the cheese is thoroughly melted and gooey.
04 - Slice the grilled sandwiches in half and serve immediately while warm.