Preheat oven to 425 degrees. In a large bowl, whisk flour, cornmeal, sugar, baking soda, salt, and pepper; make a well in center of flour mixture. Add buttermilk and eggs to well, and whisk to loosen eggs. Gently incorporate dry ingredients, then mix in cheese.
Place butter in a 9-inch cast-iron skillet; bake until butter is melted. Remove from oven, and tilt to coat bottom and sides.
Pour batter into prepared skillet; bake until golden and a toothpick inserted in center comes out clean, 20 to 25 minutes. Let cornbread cool in skillet at least 15 minutes before cutting. Serve warm or at room temperature. Wrap completely cooled bread in plastic, and store at room temperature up to 1 day.