Place the vegetable oil in a 10-inch cast-iron skillet and put the skillet in the oven. Preheat the oven to 450°F, allowing the skillet and oil to become very hot.
Meanwhile, melt the butter over medium heat. Remove from the heat and stir in the sugar.
In a large bowl, whisk together the cornmeal, flour, baking powder, baking soda, and salt.
In a separate bowl, whisk together the buttermilk and eggs. Stir in the melted butter and sugar mixture until combined.
Pour the wet ingredients into the dry ingredients and stir just until a smooth batter forms. Do not overmix.
Carefully remove the hot skillet from the oven. Pour the hot vegetable oil into the batter and stir quickly until incorporated.
Immediately pour the batter into the hot skillet. Do not stir the batter once it is in the skillet.
Return the skillet to the oven and bake at 450°F for 22–25 minutes, or until the top is golden brown and a toothpick inserted into the center comes out clean.
Carefully remove the cornbread from the skillet and transfer it to a wire rack. Let it cool for about 5–10 minutes before slicing and serving.