Preheat the oven to 400°. Put the 2 tablespoons of butter in an 8-inch-square baking dish and put the dish in the oven to heat.
Meanwhile, in a medium bowl, mix the flour with the cornmeal, sugar, baking powder and salt. In a small bowl, mix the eggs and buttermilk; add to the dry ingredients along with the melted butter and stir just until combined.
Remove the hot baking dish from the oven and tilt to spread the butter evenly. Carefully scrape the batter into the baking dish and bake for about 25 minutes, or until a toothpick inserted in the center of the corn bread comes out clean. Let cool in the baking dish for 10 minutes, then turn the corn bread out onto a wire rack to cool completely.
The corn bread can be wrapped in plastic and refrigerated for 2 days or frozen for up to 1 week.