In a medium saucepan over medium heat, combine fresh blackberries, balsamic vinegar, sugar, lemon juice, and a pinch of kosher salt. Bring this mixture to a gentle boil, then reduce the heat and allow it to simmer for about 7 to 10 minutes. Mash the berries with a fork until the jam thickens and becomes glossy. Remove from heat and let it cool completely.

While the jam cools, slice the double cream brie cheese into ¼-inch thick pieces. Arrange the slices on a plate.

Heat a skillet over medium heat and add a pat of butter. Butter one side of four slices of sourdough bread and place them butter-side down in the skillet. Layer brie slices on top of the bread in the skillet.

Spread a generous amount of blackberry jam on the unbuttered sides of the remaining slices of bread. Once the brie is melty and the bread is golden, place the jammed sides down onto the brie-covered slices. Butter the tops of these slices before flipping the sandwiches.

Once cooked, take the sandwiches out of the skillet and let them cool slightly on a cutting board. Cut each sandwich in half and serve warm.