Make the Blackberry Balsamic Jam: In a saucepan, combine blackberries, balsamic vinegar, sugar, lemon juice, and a pinch of kosher salt. Bring to a boil over medium-high heat, then reduce to a simmer. Crush the berries using a wooden spoon or potato masher.
Simmer for 7 to 10 minutes until it thickens into a jam. Let it cool after transferring to a bowl.
Prepare the Sandwiches: Slice the brie cheese into ½ inch thick pieces. Heat a skillet over medium heat; butter one side of four bread slices and place them buttered side down in the skillet.
Top with brie slices and sprinkle with chopped thyme. Let the bread fry until the brie softens and the bread turns golden.
Spread a thick layer of the blackberry jam on the unbuttered side of the remaining bread slices. Place them jam side down on the brie.
Butter the top side of the sandwiches and carefully flip. Continue frying until the cheese is fully melted and both sides of the sandwich are golden brown.
Transfer to a cutting board, let cool slightly, then cut in half to serve.