Melt butter in a small saucepan over medium heat, swirling often until lightly browned and nutty, about 4 minutes. Pour into a medium bowl, and set aside to cool.

Preheat oven to 400°F. Rub an 8-inch cast-iron skillet with the oil; place in oven to heat.

In a large bowl, combine cornmeal, flour, sugar, baking powder, salt and baking soda. Add eggs and sour cream to cooled butter, whisking well. Pour into dry ingredients, stirring just until combined. Pour batter into skillet. Bake until golden and a toothpick inserted into the center comes out almost clean, about 20 minutes. Let cool in skillet 5 minutes, then turn onto a wire rack and slice into wedges.