01 - Combine balsamic vinegar and honey in a small saucepan over medium heat. Simmer for 3 to 5 minutes, stirring occasionally, until mixture thickens. Let the glaze cool slightly.
02 - Spread butter on one side of each slice of sourdough bread. Arrange two slices, buttered side facing down, on a clean work surface.
03 - Layer brie slices on the unbuttered side of each bread base. Sprinkle shredded sharp white cheddar evenly over brie. Distribute blackberry halves across the cheese layers. Drizzle about one teaspoon of balsamic glaze over the fillings on each sandwich.
04 - Place remaining bread slices on top, buttered side facing outward, to assemble the sandwiches.
05 - Warm a nonstick skillet or griddle over medium heat. Grill sandwiches for 3 to 4 minutes per side, pressing gently, until bread turns golden and cheese becomes thoroughly melted.
06 - Transfer the sandwiches to a clean surface and let rest for one minute. Slice in half and serve immediately while warm.