In a small saucepan over medium heat, combine the blueberries, sugar, lemon juice, and lemon thyme tied bundle. Bring to a gentle boil, then reduce heat to a simmer for about 10 minutes. Continue until the mixture thickens—about another 10 minutes—then remove and discard the lemon thyme. Let the compote cool slightly.

Lay out 4 slices of sourdough bread. Spread a generous layer of the warm blueberry compote over each slice. Top each with equal parts of shredded white cheddar and mozzarella. Finish with the remaining 4 bread slices to form sandwiches.

Heat a skillet (cast iron preferred) over medium heat. Melt about 1 tablespoon of butter and place two sandwiches in the pan, pressing gently. Cook covered for 2–3 minutes per side, until the bread is golden-brown and the cheese is melted—adjust heat as needed to avoid burning.

Remove from the pan, add flaky sea salt on top if desired, and repeat with remaining butter and sandwiches. Slice in half and serve immediately.