Preheat the oven to 375 degrees F.
Place a large cast-iron skillet over medium heat and melt the butter, swirling the pan to coat the bottom and sides. Then heat the butter until it stops foaming and turns a light brown color; watch carefully so it doesn't burn. Pour the brown butter into a large bowl and set the skillet aside. Whisk the maple syrup into the butter, followed by the buttermilk and whole milk. Then whisk in the eggs. Add the cornmeal, all-purpose flour, whole-wheat flour, sugar, baking powder, salt and baking soda. Whisk together until combined.
Pour the cornmeal batter into the cast-iron skillet. Bake until firm in the center, golden brown on the sides, and an inserted toothpick comes out clean, about 20 minutes. Let cool slightly, then serve warm with butter.