For the filling: Butter the sides and bottom of the insert of a 6-quart slow cooker. Whisk together the cornstarch, lemon juice and 2 tablespoons water in a small bowl. Add the mixture to the insert, add the berries and granulated sugar and toss.
For the biscuit topping: Whisk together the flour, 2 tablespoons of the granulated sugar, baking powder and salt in a large bowl. Add the butter and rub the mixture together with your fingers so that it forms a coarse meal with some pea-sized lumps. Add the milk and stir with a wooden spoon to form a wet dough.
Drop spoonfuls of the dough on top of the berry mixture. Combine the remaining 1 tablespoon granulated sugar and the cinnamon and sprinkle over the dough. Cover the slow cooker and set to high for 3 to 4 hours. The cobbler should be bubbly and the biscuits should be cooked through. Let rest for 20 minutes before serving.
For the sour cream topping: Meanwhile, combine the heavy cream, sour cream and confectioners’ sugar in a medium bowl and beat with an electric mixture until the mixture forms medium peaks.
Spoon the cobbler into bowls and top with the whipped sour cream topping.