Apple and Blackberry Polenta Cobbler

Name: Apple and Blackberry Polenta Cobbler
Ingredients: 3 lb baking apples (such as Honeycrisp or Pink Lady), peeled, cored, cut into ¼” wedges; 1 10-oz. package frozen blackberries (about 2 cups), thawed; ⅓ cup sugar; 3 tablespoon cornstarch; 3 tablespoon honey; 2 tablespoon unsalted butter, cut into ½” pieces; 2 tablespoon fresh lemon juice; ½ cup medium-grind polenta or cornmeal (not quick-cooking); 2 teaspoon baking powder; ¾ teaspoon kosher salt; 1½ cups all-purpose flour plus more for work surface; ¼ cup sugar plus more for sprinkling; 6 tablespoon (¾ stick) chilled unsalted butter, cut into ½” pieces; 1 cup heavy cream plus more for brushing