Peel, core, and thinly slice the apple. Melt 2 tablespoons butter in a 9-inch, cast-iron skillet over medium-low heat. Add apple slices and sauté until wilted, about 3 minutes. Transfer slices to a plate. Increase heat to high, add 1/4 cup sugar to skillet and cook, stirring frequently, until sugar melts and turns golden, about 3 minutes. Remove skillet from heat and arrange the apple slices in a circular pattern over the bottom. Set skillet aside.
Chop pecans, toss with 3 tablespoons sugar and the cinnamon, and set aside. Sift flour with baking powder, baking soda, and salt; set aside. Beat remaining butter in a bowl until lightened. Gradually beat in 1 cup sugar, eggs, one at a time, sour cream, and vanilla. Fold dry ingredients into the batter.
Adjust oven rack to middle position and heat oven to 350°F. Sprinkle half of the pecan mixture over the apples. Using your fingers or a spoon, carefully spread half of the cake batter over pecans. Sprinkle the remaining pecan mixture over the batter and carefully spread the remaining cake batter over pecans. Bake the cake until top is golden and a toothpick inserted in the center comes out clean, about 45 minutes. Cool on a rack for 5 minutes. Run a small knife along the edge of the cake and carefully invert cake onto a serving platter. If apples stick to the skillet, loosen with a knife and arrange on the cake. Serve slightly warm or at room temperature. (Can cover and store at room temperature up to 2 days.)
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