In a medium saucepan over medium heat, combine fresh blueberries, sugar, lemon juice, and lemon thyme. Stir to combine and bring to a gentle boil for about 5 minutes. Reduce heat to low and simmer for 20 minutes, stirring occasionally, until thickened. Remove from heat and cool completely.

While the compote cools, lay out slices of bread. Spread blueberry compote on one side of each slice. Layer white cheddar and mozzarella over the compote, then top with the remaining slices.

Heat a non-stick skillet over medium heat and add unsalted butter. Once melted, grill two sandwiches for 2-3 minutes until golden brown. Flip and grill the other side for another 2-3 minutes until cheese is melted and both sides are golden.

Remove sandwiches from skillet, let rest for a minute, then cut in half. Optionally sprinkle with flaky sea salt and fresh thyme before serving.