Preheat oven to 400. Place butter in 10 1/2-inch cast-iron skillet and heat in oven 10 to 15 minutes.
In a bowl, whisk together flour, cornmeal, baking powder, baking soda and salt. Stir in 1 cup of the cheese.
Remove skillet from the oven and pour melted butter into the flour mixture. Add buttermilk and eggs and stir to combine. Pour batter back into prepared skillet and smooth the top. Bake until a toothpick inserted in the center comes out clean and the bread is golden, about 20 minutes.
Just before cornbread is done, remove from the oven and sprinkle with remaining 1/4 cup of cheese. Bake until cheese is melted, 3 to 5 minutes. transfer to a rack to cool slightly before serving.