In a medium saucepan, combine fresh blueberries, white sugar, lemon juice, and lemon thyme. Cook over medium-high heat, stirring occasionally until the mixture reaches a boil. Once boiling, reduce the heat to a gentle simmer and let it cook for about 20 minutes, or until the mixture thickens and the blueberries burst.

While the blueberry compote cools, take four slices of sturdy bread. Spread a generous layer of the cooled blueberry compote on two slices. Sprinkle shredded white cheddar and whole milk mozzarella cheese over the compote. Top with remaining slices to form the sandwiches, pressing gently to secure layers.

Heat a skillet over medium heat and add a pat of butter. Place the sandwiches in the skillet and cover with a lid. Cook for 2-3 minutes until the bottom is golden brown. Carefully flip each sandwich and cook the other side for another 2-3 minutes.

Once your crispy blueberry grilled cheese sandwiches are golden and the cheese is gooey, remove them from the skillet. Slice each sandwich diagonally and serve immediately, enjoying the contrast of flavors and textures.