Make Blueberry Compote: In a medium saucepan, combine fresh blueberries, granulated sugar, lemon juice, and lemon thyme over medium heat. Bring to a gentle boil, then reduce heat to low and simmer for 15-20 minutes, stirring occasionally until thickened.

Prepare the Sandwiches: Spread the cooled blueberry compote evenly on each slice of bread. Sprinkle grated white cheddar and mozzarella cheese over the compote on two slices, then top with the remaining slices to form sandwiches.

Cook the Sandwiches: Melt a tablespoon of unsalted butter in a skillet over medium heat. Place the sandwiches in the skillet, cooking for 2-3 minutes on one side until golden brown.

Flip and Finish Cooking: Flip the sandwiches, adding another tablespoon of butter as needed. Cook for an additional 2-3 minutes until the second side is golden brown and the cheese is melted.

Serve: Let the sandwiches cool for a minute, cut in half, and optionally sprinkle with flaky sea salt and fresh thyme before serving.