Stir together the flour, cornmeal, baking powder, baking soda, and salt in a medium bowl. Set aside.
Preheat the oven to 350°. Butter a 9-inch deep dish pie plate, and set it aside. Beat ½ cup butter and brown sugar until fluffy, about 3 minutes. Add the egg and vanilla, and beat until just combined. Add half of the flour mixture, the buttermilk, then the remaining flour, beating until combined after each addition.
Spoon the batter into the prepared pie plate; spread in an even layer using the back of a spoon or a spatula. Scatter the berries evenly over the top of the batter; do not press them down. Sprinkle the berries and batter with 2 tablespoons turbinado sugar, if using.
Bake until the top is golden brown and a wooden pick inserted into the cake (not a strawberry!) comes out clean, 40 to 45 minutes. Let cool before dusting with powdered sugar and serving with more sliced strawberries, if you like.
