This version is for a cast iron skillet. Heat oven to 350.

FOR THE TOPPING: Melt butter in 9 or 10-inch skillet. Add brown sugar and stir until bubbly. Turn down to low and let it simmer. Watch the color so it doesnt burn. Add rum at the end, carefully, because it can bubble up.

Slice the peeled apples into 1/4-inch-thick slices across the equator then slice down so you have crescents. Toss with lemon juice. Arrange apple slices in skillet on brown sugar mixture in about 3 layers. Press down with spatula to get them down into the syrup. Let this simmer on low while you make the cake part...6-8 minutes. You dont want to cook the apples through, just warm them up

FOR THE CAKE: Whisk flour, cornmeal, baking powder, and salt together in medium bowl; set aside. Whisk granulated sugar, brown sugar, and eggs together in large bowl until thick and homogeneous, about 45 seconds. Slowly whisk in butter until combined. Add sour cream and vanilla; whisk until combined. Add flour mixture and whisk until just combined. Pour batter into pan and spread evenly over fruit. Bake until cake is golden brown and toothpick inserted into center comes out clean, 35 to 40 minutes.

Cool pan on wire rack at least 20 minutes. Run paring knife around sides of cake to loosen. Briefly place skillet over medium heat to re-melt the caramel syrup so it pops out better. Place large plate over skillet and holding tightly, invert and slowly lift skillet off plate. If any fruit sticks to pan bottom, remove and position it on top of cake. Let cake cool 20 minutes (or longer to cool it completely), then transfer to serving platter, cut into pieces, and serve.