Preheat the oven to 350°F. Place a rack in the middle of the oven.
In a food processor, whiz the flour, sugar, baking powder and salt for biscuits. Cut in the cold butter and process until it forms a course meal. Add the buttermilk slowly until the dough comes together.
In a large bowl, mix blackberries, rhubarb, sugar, cornstarch, lemon juice and zest.
Pour fruit mix into a greased 9x9-inch baking pan or 9-inch pie pan. Pat handfuls of dough into 1/4 to 1/2-inch thick, palm-sized disks. Lay the dough over the fruit filling. Brush the tops of the biscuits with the remaining tablespoon of buttermilk and sprinkle with coarse sugar.
Place the cobbler on a baking sheet in the oven. Bake until the biscuits are golden brown around the edges and the fruit filling is bubbling, 45 to 50 minutes.
Meanwhile, in a bowl of a stand mixer or using a hand mixer, beat the cream cheese until smooth. Add the milk, confectioners’ sugar and vanilla extract and beat until smooth. Set aside.
Let the cobbler cool and then drizzle the cream cheese glaze over top. Serve immedatetly with extra glaze on the side for dipping.
