Combine fresh blueberries, sugar, lemon juice, and lemon thyme in a saucepan over medium heat. Cook until the blueberries soften and release their juices, about 5 minutes.

Lay out four slices of sourdough bread. Sprinkle half of the grated white cheddar and shredded mozzarella over each slice. Spoon some blueberry mixture onto two slices and top with another slice of bread.

In a large skillet over medium heat, melt two tablespoons of butter. Add the assembled sandwiches and grill for about 3–4 minutes on each side until golden brown and crispy.

Once both sides are crisped and the cheese has melted, remove from the skillet. Allow to cool slightly before cutting in half. Sprinkle with flaky sea salt and serve warm.