Prepare the Blueberry Lemon Thyme Jam: In a 2-quart saucepan, combine the fresh blueberries, white sugar, lemon juice, and the bundled lemon thyme. Place the pan over medium-high heat and bring the mixture to a boil. Once boiling, reduce the heat to medium-low and simmer for about 10 minutes. Lower the heat further to low and continue to simmer for another 10 minutes, stirring frequently, until the jam has thickened into a jammy consistency. Remove the lemon thyme bundle and let the jam cool for at least 10 minutes.

Assemble the Sandwiches: Lay out four slices of sourdough bread on a clean surface. Spread a generous, even layer of the cooled blueberry jam onto each of the four slices. Sprinkle the shredded white cheddar and mozzarella cheeses over the jam. Place the remaining four slices of bread on top to close the sandwiches.

Grill the Sandwiches: Heat a 12-inch cast iron skillet over medium heat and melt 2 tablespoons of butter. Place two of the assembled sandwiches into the skillet and cover with a lid. Cook for 2 to 3 minutes until the bottom is golden brown. Flip the sandwiches and cook for another 2 to 3 minutes until the second side is crispy and the cheese has fully melted. Remove these sandwiches to a cutting board, add the remaining 2 tablespoons of butter to the skillet, and repeat the process for the remaining two sandwiches.

Finish and Serve: Once all sandwiches are cooked, cut them in half. Sprinkle the tops with a touch of flaky sea salt and a few fresh lemon thyme leaves for an aromatic finish. Serve while hot and melty.