Adjust oven rack to center position and heat oven to 425 degrees. Grease a 9" square metal pan.
Whisk cornmeal, flour, baking powder, baking soda, sugar, and salt together in large bowl. Push dry ingredients up side of bowl to make a well.
Crack eggs into well and stir lightly awith wooden spoon, then add buttermilk and milk. Stir wet and dry ingredients quickly until almost combined. Add melted butter; stir until ingredients are just combined.
Pour batter into greased pan. Bake until top is golden brown an dlightly cracked, about 25 minutes.
Transfer pan to wire rack to cool slightly and serve warm.