In a medium saucepan over medium heat, combine 2 cups of fresh blueberries, 1/4 cup of sugar, 1 tablespoon of lemon juice, and a few sprigs of lemon thyme. Stir gently and bring to a boil, allowing the blueberries to release their juices, about 5 minutes.

Once boiling, reduce heat to a simmer and cook the mixture for 20 minutes, stirring frequently until thickened. Remove from heat and let cool.

Spread a generous tablespoon of the cooled blueberry compote on 4 slices of bread. Layer a mix of shredded white cheddar and mozzarella cheese, then top with remaining bread to form 4 sandwiches.

Heat a large skillet over medium-low heat, add 1 tablespoon of butter, and spread it evenly. Grill the assembled sandwiches for about 3 minutes per side until golden-brown and cheese is melted.

Remove the sandwiches from the skillet, let rest for a minute, slice in half, and optionally sprinkle with flaky sea salt and fresh thyme before serving.