Prepare the Blueberry Compote: In a medium saucepan, combine fresh blueberries, sugar, lemon juice, and lemon thyme. Bring to a boil over medium-high heat, then reduce heat and let simmer for 20 minutes until thick.

Assemble the Sandwiches: Spread cooled blueberry compote on slices of bread. Sprinkle with white cheddar and mozzarella, top with another slice of bread, and press gently.

Grill the Sandwiches: Heat skillet over medium heat, add butter, and once sizzling, place sandwiches in the skillet. Grill each side for 2-3 minutes until golden brown.

Flip and Finish Grilling: Carefully flip to grill the other side, adding more butter if necessary. Grill until both sides are golden and cheese is melted.

Serve and Enjoy: Remove from skillet, let cool for a minute, slice in half, and serve warm, optionally with flaky sea salt and a side.