Make topping: Peel, core, and quarter apples.
In a well seasoned 10 1/4 by 2-inch cast iron skillet melt butter over moderately low heat just until melted (butter should not separate). Stir in sugar until combined well. Arrange apple quarters decoratively, cut sides up, in skillet and sprinkle walnuts evenly between apples. Cook mixture, undisturbed, 25 to 35 minutes, or until the apples are tender in centers and sugar is a golden caramel.
Preheat oven to 375 degrees.
Make cake batter while topping is cooking: Peel apple and chop fine.
In a bowl whisk together flour, baking powder, salt, and cinnamon.
In another bowl with an electric mixture beat butter and sugar until light and fluffy. Beat in vanilla and gingerroot and add eggs, 1 at a time, beating well after each addition. Beat in sour cream and with mixer on low speed beat in flour mixture gradually until just combined. Fold chopped apple into batter.
Remove skillet from heat and spoon batter evenly over topping. With a metal spatula spread batter (being careful not to disturb topping), leaving a 1/4-inch border of cooked apples uncovered. If using a non-stick stick skillet with a plastic handle, wrap handle in a double thickness of foil. Put skillet in a shallow baking pan and bake cake in middle of oven 25 to 35 minutes, or until a tester comes out with crumbs adhering and cake is golden brown. Cool cake in skillet on a rack 10 minutes. Run a thin knife around edge of skillet and carefully invert cake onto a plate.
Serve cake warm or at room temperature with caramel sauce and whipped cream.