Start by combining blackberries, balsamic vinegar, sugar, lemon juice, and salt in your saucepan. Cook over medium heat, stirring occasionally, until berries break down completely. The mixture should thicken slightly and become jammy in consistency, taking about 8-10 minutes total cooking time.
Spread butter generously on one side of each bread slice, ensuring complete coverage for even browning. Place brie cheese slices on the unbuttered side of four bread pieces. Spoon warm blackberry mixture over the cheese, then sprinkle with fresh thyme before topping with remaining bread slices.
Heat your skillet over medium heat until it's evenly warmed but not smoking hot. Place sandwiches buttered-side down and cook for 3-4 minutes until golden brown. Flip carefully using a spatula, then cook another 3-4 minutes until the second side browns and cheese melts completely.