Preheat the oven to 375˚ F. Butter a 9-inch cast-iron or other ovenproof skillet. Whisk the cornmeal, flour, sugar, baking powder, salt and baking soda in a large bowl. In a separate large bowl, whisk the eggs, buttermilk, milk and melted butter until smooth. Whisk the buttermilk mixture into the cornmeal mixture and pour the batter into the prepared skillet; smooth the top. Bake until golden and a toothpick inserted into the center comes out clean, 25 to 30 minutes.