1. Preheat oven to 350°. Coat a 9-inch round cake pan with cooking spray.
2. To prepare topping, combine 3/4 cup sugar and 1/4 cup water in a small heavy saucepan over medium-high heat; cook until sugar dissolves, stirring gently as needed to dissolve sugar evenly (about 3 minutes). Continue cooking for 4 minutes or until golden (do not stir). Immediately pour into prepared cake pan, tipping quickly to coat bottom of pan. Arrange apple slices in concentric circles in pan over the warm caramel. Sprinkle with nuts; set aside.
3. To prepare cake, weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking powder, and salt; stir with a whisk.
4. Combine 2/3 cup sugar and butter in a large bowl; beat with a mixer at medium speed until light and fluffy. Add egg yolks and vanilla to sugar mixture; beat until combined. Add flour mixture and milk alternately to sugar mixture, beginning and ending with flour mixture; mix after each addition.
5. Place egg whites in a large, clean bowl. Beat egg whites with mixer at high speed until stiff peaks form using clean, dry beaters. Gently fold egg whites into batter. Spread batter over apples. Bake at 350° for 35 minutes or until a wooden pick inserted in center comes out clean. Cool on a wire rack in pan for 5 minutes. Loosen edges of cake with a knife; invert cake onto a serving plate. Serve warm or at room temperature.