How to Make It
Preheat the oven to 350°. Lightly butter a 9-inch square baking pan. In a medium bowl, whisk 1 1/4 cups of the flour with the cornmeal, granulated sugar, baking powder, cinnamon, baking soda, salt and nutmeg.
In an extra-large skillet, melt 4 tablespoons of the butter over moderately high heat. Add the apples and brown sugar and cook, stirring occasionally, until the apples are just tender and the sugar is beginning to caramelize, about 12 minutes. Stir in the apple cider, lemon juice and vanilla and cook until the liquid has reduced by half, about 5 minutes. Remove the skillet from the heat and stir in the remaining 1 tablespoon of flour until just combined.
Scrape the apple filling into the prepared pan and, using a rubber spatula, pat into an even layer. Sprinkle the dry ingredients over the apple filling. Dot the remaining 1 1/2 sticks of butter over the dry ingredients. Bake the cake, turning the pan from back to front halfway through, for about 50 minutes, until the top is golden brown and the filling is bubbling. Transfer the cake to a wire rack and let cool slightly, about 15 minutes.
Spoon the cake onto plates and serve with ice cream.