Heat oven to 425 degrees. Grease a 9-inch square baking pan with butter
Whisk the cornmeal, flour, baking powder, baking soda, sugar, and salt together. Push the dry ingredients up the sides of the bowl to make a well
Crack the eggs into the well and stir lightly with a wooden spoon, then add the buttermilk & milk. Stir the wet & dry ingredients quickly until almost combined. Add the melted butter and stir until the ingredients are just combined
Pour the batter into the greased pan. Bake until the top of the cornbread is golden brown and lightly cracked and the edges have pulled away from the sides of the pan, about 25 min
Transfer the pan to a wire rack to cool slightly, 5 to 10 minutes. Cut the cornbread into squares and serve warm. (Pan can be wrapped in foil and stored at room temperature up to 1 day). Reheat cornbread in a 350° oven for 10-15 min)