In a saucepan over medium heat, combine fresh blueberries, sugar, lemon juice, and lemon thyme. Stir gently and bring to a boil, then reduce the heat and simmer uncovered for about 20 minutes, until thickened.

Take two slices of your choice of sturdy bread and spread a layer of cooled blueberry compote on one side of each slice. Evenly sprinkle the grated white cheddar and mozzarella cheese over the compote.

In a skillet, melt a tablespoon of butter over medium heat. Carefully place the assembled sandwiches in the skillet, cooking for 2-3 minutes until golden brown.

Gently flip the sandwiches and add another tablespoon of butter if needed. Cook for an additional 2-3 minutes, pressing slightly, until both sides are crispy.

Remove the sandwiches from the skillet, let them rest for a minute, and slice in half to reveal the gooey cheese and blueberry compote inside.