In a medium saucepan, combine fresh blueberries, a dash of white sugar, lemon juice, and lemon thyme over medium heat. Bring to a gentle boil, then reduce heat to low and simmer for about 20 minutes. Stir occasionally until thickened.

Spread a generous layer of blueberry compote on one side of each slice of bread. Top one slice with an even mix of shredded white cheddar and mozzarella, then place the second slice on top, compote side down.

Heat a skillet over medium heat and add a tablespoon of unsalted butter. Once bubbling, place the sandwiches in the skillet and grill for 2-3 minutes until golden brown. Flip and grill the other side, pressing down gently.

After about 2-3 minutes on the second side, check for melty cheese and a crispy exterior. Remove from the skillet, let rest, slice diagonally, sprinkle with salt and thyme, and serve immediately.