In a medium saucepan, combine fresh blueberries, sugar, lemon juice, and lemon thyme. Stir over medium heat until boiling, then reduce heat and let simmer for 20 minutes until thickened.

Spread blueberry compote on one side of four slices of sturdy bread. Layer white cheddar and mozzarella on two of the slices. Place the remaining slices on top, compote side down.

Preheat a skillet over medium-low heat and add butter. Once melted, place sandwiches in the skillet and grill for 2-3 minutes per side until golden brown and cheese is melted.

Remove from skillet, let rest for a minute, then cut in half. Sprinkle with flaky sea salt and fresh thyme if desired.