In a large bowl, combine the blueberries, blackberries, cornstarch and 2 tablespoons sugar. Transfer to a 13x9-in. baking dish coated with cooking spray.
In a large bowl, cream butter and remaining 1/3 cup sugar until light and fluffy, 5-7 minutes. Beat in lemon zest. Combine the flour, cornmeal, baking powder and salt; add to the creamed mixture alternately with sour cream. Beat just until combined. Drop by tablespoonfuls onto berry mixture.
In a small bowl, combine the coarse sugar, lemon juice and honey; brush over dough. Bake at 350° until filling is bubbly and a toothpick inserted in topping comes out clean, 35-40 minutes. Serve warm.
Test Kitchen tips
This rustic cobbler isn't overly sweet, so it lets the berries really shine.
Consider serving this dessert with a scoop of vanilla bean ice cream or lightly sweetened whipped cream.