In a medium saucepan, combine fresh blueberries, white sugar, lemon juice, and lemon thyme. Heat over medium-high until the mixture comes to a rolling boil, then reduce to a simmer. Stir occasionally and cook for about 15 to 20 minutes, or until the compote thickens and coats the back of a spoon. Allow it to cool completely before using.

Take your choice of sturdy bread and spread a generous layer of blueberry compote on one side of each slice. Layer sharp white cheddar and creamy mozzarella atop the compote. Place the remaining slices of bread on top, compote side down.

Preheat a cast iron skillet or non-stick frying pan over medium heat. Once hot, add butter and let it melt. Carefully place the assembled sandwiches in the pan. Grill for about 2 to 3 minutes on each side, using a lid to cover the pan.

When the bread is crispy and golden, and the cheese is melted, remove the sandwiches from the skillet. Allow them to cool for a moment before cutting them in half. Optionally, sprinkle with flaky sea salt and garnish with fresh thyme. Serve warm.