Apple Blackberry Crisp

Name: Apple Blackberry Crisp
Ingredients: 1/2 cup unsalted butter, cold and cut into 1/4-inch cubes, plus more for the pan; 3/4 cup all-purpose flour; 2/3 cup dark brown sugar; 1/2 cup quick oats; 1 teaspoon cinnamon; 3/4 teaspoon baking powder; 1/4 teaspoon allspice; 1/4 teaspoon ground ginger; 1/4 teaspoon kosher salt; 2 pounds baking apples (about 6); 1/2 pound blackberries (fresh is ideal, but frozen works too); 1/4 cup granulated sugar; 1 tablespoon cornstarch; 1 tablespoon fresh lemon juice