Place a 10-inch cast-iron skillet on the center rack and preheat the oven to 400 degrees F. Brush the inside of the hot skillet with the melted butter, coating the bottom and sides generously.

Cut the stick of butter into pieces and melt it in a light-colored saucepan over medium heat. Watch the color change as it cooks.

Keep cooking, swirling the pan often, as the butter foams, then the foam subsides and the milk solids sink and turn amber, about 5 to 6 minutes. Pull it off the heat the moment it smells nutty, then pour it into a heatproof bowl so it stops cooking.

Whisk the maple syrup straight into the warm brown butter, then let the mixture cool for 5 minutes so it does not scramble the eggs.

In a large mixing bowl, whisk together the cornmeal, flour, maple sugar, baking powder, baking soda, and salt until evenly combined and free of lumps.

In a second bowl, whisk the buttermilk and eggs until smooth, then whisk in the cooled brown butter and maple mixture.

Pour the wet mixture into the dry ingredients and fold gently with a spatula just until no dry streaks remain. A few small lumps are fine; overmixing makes the crumb tough.

Carefully pour the batter into the hot greased skillet; it should sizzle faintly at the edges. Smooth the top with the spatula and slide it into the oven.

Bake for 20 to 25 minutes, until the top is deep golden and a toothpick inserted in the center comes out clean or with a few moist crumbs.

Cool in the skillet for 10 minutes, then cut into 8 wedges and serve warm with softened butter and extra maple syrup.