In a medium saucepan over medium heat, combine 2 cups of fresh blueberries, ¼ cup of white sugar, 1 tablespoon of lemon juice, and 1 teaspoon of lemon thyme. Bring the mixture to a boil, then reduce the heat and simmer for about 10 minutes, stirring occasionally, until the blueberries have softened and the compote thickens. Remove from heat and let it cool slightly.

Once the blueberry compote has thickened, set it aside to cool for about 5–10 minutes.

Take four slices of thick bread and lay them flat on a clean surface. Generously spread the cooled blueberry compote on two of the slices, then layer with 2 ounces each of shredded white cheddar and mozzarella cheese. Place the remaining bread slices on top to form two complete sandwiches, ensuring the filling is evenly distributed.

Preheat a large skillet over medium heat and add 2 tablespoons of butter, allowing it to melt and coat the pan. Carefully place the assembled sandwiches into the skillet, cooking for about 3–4 minutes on each side.

Once both sides of the sandwiches are golden and the cheese is fully melted, remove them from the skillet. Let them cool for a minute, then slice each sandwich in half diagonally.