Preheat the oven to 400 degrees Fahrenheit.
Rinse blackberries under cool water, and pat until just slightly damp. Add the blackberries to a casserole dish. Sprinkle the blackberries with brown sugar, cornstarch and lemon juice. Toss to coat the berries.
In a separate bowl add the flour, sugar, baking powder and salt. Use a pastry cutter or two knives to cut in the cold butter. The mixture will be crumbly.
Stir in the heavy cream, sour cream and vanilla extract. The mixture will be very thick, and need to be pushed together with the back of a spoon.
Drop spoonfuls of dough on top of the blackberries. There likely will not be enough to cover all of the blackberries and this is perfectly fine, it actually looks even better that way.
Bake for 35-40 minutes or until the cobbler is golden brown and the berries are bubbling around the edges.
Allow to cool slightly, then serve warm. Vanilla ice cream makes the perfect topping.