Adjust oven rack to center position and preheat oven to 425°F. Place butter in a 10-inch cast iron skillet. Transfer skillet to oven and heat until the butter is melted and well browned, 10- 12 minutes. Once browned, pour butter into a heatproof cup or bowl, leaving about 1 tablespoon remaining in the skillet.
Combine cornmeal, flour, sugar, salt, baking powder, and baking soda in a large bowl. Combine eggs, sour cream, and buttermilk in a second bowl and whisk until well combines and smooth. Whisking constantly, slowly drizzle in the reserved brown butter. Whisk wet ingredients into dry ingredients until well combined.
Remove skillet from the oven and swirl to coat all surfaces with melted butter. Spoon batter into skillet, smooth top lightly and then transfer to oven. Bake until lightly golden brown on top and a wooden skewer inserted into cornbread comes out with no crumbs, about 20 minutes. Let cool 10 minutes in skillet. Serve with extra butter at the table.
Notes
**Tip – Leftover cornbread can be stored in a sealed container in the refrigerator for up to 3 days.**