01 - Combine balsamic vinegar and honey in a small saucepan over medium heat. Simmer gently for 3 to 5 minutes, stirring occasionally, until the mixture thickens. Remove from heat and allow to cool slightly.
02 - Spread unsalted butter evenly on one side of each slice of sourdough bread. Place two slices, buttered side down, on a clean surface.
03 - Arrange brie cheese slices over the bread, then sprinkle with shredded sharp white cheddar and evenly distribute halved blackberries. Drizzle each portion with 1 teaspoon of prepared balsamic glaze.
04 - Top each assembly with another slice of bread, buttered side facing up.
05 - Preheat a nonstick skillet or griddle over medium heat. Place sandwiches in the skillet and cook for 3 to 4 minutes per side, pressing gently, until the bread is golden and cheese is melted.
06 - Transfer grilled sandwiches to a cutting board. Allow them to rest for 1 minute, then slice and serve warm.