01 - Combine balsamic vinegar and honey in a small saucepan over medium heat. Simmer for 3 to 5 minutes, stirring occasionally, until the mixture reduces and thickens slightly. Set aside to cool.
02 - Spread softened butter on one side of each sourdough bread slice. Lay two slices, buttered side down, on a clean surface.
03 - Evenly layer brie slices on the bread, followed by the shredded white cheddar and halved blackberries. Drizzle each sandwich with one teaspoon of balsamic glaze.
04 - Top with remaining bread slices, buttered side up, to form sandwiches.
05 - Preheat a nonstick skillet or griddle over medium heat. Place sandwiches on the skillet and cook for 3 to 4 minutes per side, pressing gently with a spatula, until the bread is golden-brown and the cheese is melted.
06 - Transfer sandwiches to a cutting board. Let rest for 1 minute, then slice and serve immediately.