Preheat the oven to 400 degrees . Grease an 8-inch square baking dish. In a large bowl, whisk together the flour, cornmeal, baking powder, baking soda and salt; set aside. In a separate bowl, whisk the buttermilk, eggs and brown sugar.
Using a rubber spatula, make a well in the center of the dry ingredients. Pour in the wet ingredients and stir together until just combined. Fold in the butter until just incorporated. Pour the batter into the prepared dish and bake until a toothpick inserted in the center comes out clean, about 30 minutes.