Preheat the oven to 350° F. Line a baking sheet with parchment paper.
Place the flour, butter, cheese and a few grinds of black pepper in the bowl of a food processor. Process until the dough just comes together and starts to form a ball.
Dump the dough onto a lightly floured surface and knead a few times to pull the dough together. Roll out to 1/4 inch thick with a floured rolling pin. Cut rounds out of the dough with a floured 1-inch cutter and transfer the rounds to the parchment-lined baking sheet.
Using the back of a round half-teaspoon or your knuckle, make an indentation in the top of each dough round. Spoon about 1/4 teaspoon of blackberry preserves into each indentation, using your finger to push the preserves as best as possible into the indentations.
Bake the savories for 10 to 14 minutes, until the preserves are bubbling and the pastry is light golden on the bottom.
Let cool on the baking sheet for at least 10 minutes, then remove to a wire rack to cool. Serve warm, or make a day ahead and keep them in two layers separated by waxed paper in an airtight container.