In a saucepan over medium heat, combine fresh blueberries, granulated sugar, lemon juice, and fresh lemon thyme. Bring to a gentle boil, then simmer for about 20 minutes until thickened.

While the compote cools, spread a generous layer of blueberry compote on one side of each bread slice, then layer sharp white cheddar and mozzarella cheese over the compote. Top with another slice of bread, compote side down.

Heat a skillet over medium-low heat and melt one tablespoon of butter. Cook the sandwiches for 2-3 minutes until golden brown, then flip and add more butter if needed, cooking for another 2-3 minutes until cheese melts.

Once grilled, remove the sandwiches from the skillet, let cool for a minute, cut in half, and sprinkle with flaky sea salt and additional lemon thyme if desired. Serve warm.