Prepare the Blackberry Balsamic Jam:
In a saucepan, combine blackberries, balsamic vinegar, sugar, lemon juice, and a pinch of kosher salt.
Bring the mixture to a boil over medium-high heat, then reduce the heat to low.
Simmer for 7-10 minutes, breaking down the blackberries with a wooden spoon or potato masher until the mixture thickens into a jam.
Transfer to a small bowl and allow to cool.
Assemble the Grilled Cheese:
Slice the brie into ½ inch thick slices.
Heat a skillet over medium heat. Spread butter on one side of 4 slices of sourdough bread.
Place the buttered side down in the pan, top with slices of brie, and allow the cheese to soften while the bread turns golden brown.
Sprinkle fresh thyme over the melted brie.
Add the Jam and Grill:
Spread a thick layer of the blackberry balsamic jam on the unbuttered side of the remaining slices of bread.
Place the jam-covered side down onto the brie-topped slices in the pan.
Butter the top side of the bread and carefully flip the sandwich.
Cook until the second side is golden brown and the cheese is fully melted.
Serve:
Transfer the sandwiches to a cutting board and allow them to cool for a few minutes.
Slice in half and serve warm.