On the stove, melt the brown butter over medium heat in a 11- or 12-inch cast iron skillet (must be ovenproof). Swirl pan to lightly coat sides and bottom.
Once melted, pour butter into a large bowl. Do not wipe pan. Whisk the maple syrup into the butter, then whisk in the buttermilk. Once mixture is cool to the touch, whisk in the eggs. Then, mix in the cornmeal, flour, baking powder, salt, and baking soda.
If skillet is no longer hot, reheat it briefly on the stove for a few minutes. Pour batter back into skillet. Bake until the top is golden brown and a toothpick inserted emerges clean, 30-40 minutes. Cool in the skillet for 10 minutes before slicing.