Combine fresh blueberries, white sugar, lemon juice, and bundled lemon thyme in a 2-quart saucepan over medium-high heat. Bring to a boil, then reduce heat to medium-low and simmer for 10 minutes. The berries will start breaking down and releasing their juices.

Lower heat to low and continue simmering for another 10 minutes, stirring often, until the jam is thickened and coats the back of a spoon. The jam should hold its shape when dragged through with a spoon. Remove and discard the lemon thyme bundle. Allow to cool to room temperature, about 20-25 minutes total.

Lay out 4 slices of bread and spread the cooled blueberry jam evenly onto each slice, going almost to the edges. Sprinkle the white cheddar and mozzarella cheeses over the jam, distributing evenly. Top with the remaining 4 bread slices to form sandwiches and press down gently.

Heat a 12-inch cast iron skillet over medium heat and add 2 tablespoons of butter, letting it melt. Place two assembled sandwiches in the skillet and cover with a lid. Cook for 2-3 minutes until golden on the bottom, then flip and cook for another 2-3 minutes until the bread is golden and the cheese is melted. If bread browns too quickly, reduce heat to medium-low.

Transfer the cooked sandwiches to a cutting board, add the remaining 2 tablespoons of butter to the skillet, and repeat the process with the remaining two sandwiches. Cut each sandwich in half diagonally and sprinkle with flaky sea salt before serving.