Make the Blackberry Balsamic Jam

1. Step 1: In a medium saucepan over medium-high heat, combine fresh blackberries, balsamic vinegar, sugar, lemon juice, and a pinch of salt.
2. Step 2: Stir occasionally as the mixture comes to a boil, then reduce the heat to a simmer.
3. Step 3: Use a wooden spoon or a potato masher to gently crush the blackberries, releasing their juices and helping them break down.
4. Step 4: Allow the mixture to simmer for 7-10 minutes, stirring often, until it thickens to a jam-like consistency.
5. Step 5: Remove the saucepan from the heat, transfer the jam to a bowl, and let it cool completely.

Prepare the Sandwiches

1. Step 1: Slice the brie cheese into approximately ½-inch thick pieces.
2. Step 2: Heat a skillet over medium heat. While the skillet warms up, butter one side of each of the four slices of sourdough bread.
3. Step 3: Place the buttered slices, buttered side down, in the skillet. Arrange the brie slices on top of the bread, covering each slice generously.
4. Step 4: Sprinkle the chopped fresh thyme over the brie cheese. Let the bread cook in the skillet until it becomes golden brown and the cheese begins to soften slightly.

Assemble the Grilled Cheese

1. Step 1: Spread a generous layer of the cooled blackberry balsamic jam on the unbuttered side of the remaining four slices of sourdough bread.
2. Step 2: Carefully place each jam-covered slice (jam side down) on top of the brie-topped bread slices in the skillet to form sandwiches.
3. Step 3: Butter the exposed top of each sandwich. Gently flip each sandwich, allowing the other side to cook until it becomes golden brown and the brie is fully melted.
4. Step 4: Once both sides of the sandwich are golden and crispy, transfer them to a cutting board.
5. Step 5: Let the sandwiches cool slightly, then cut each one in half. Serve while warm for the best flavor and texture.