Stir together brown sugar and 4 tablespoons butter in a small bowl. Scrape seeds from 1 vanilla bean, or pour 1 1/2 teaspoons extract into mixture; stir until smooth. Spread mixture in an even layer onto bottom of a 10-inch cast-iron or ovenproof skillet. Arrange apple wedges over sugar mixture, overlapping slightly in a circular pattern; fill in center of skillet with remaining wedges. Spray sides of skillet with cooking spray.
Heat oven to 350°.
Combine almond paste, remaining 6 tablespoons butter, and granulated sugar in a large mixing bowl; scrape seeds from 1/2 vanilla bean, or pour 3/4 teaspoon extract into mixture. Beat with electric mixer on medium speed about 2 minutes or until creamy and smooth. Beat in eggs, 1 at a time, until incorporated. Beat in flour, baking powder, and salt until combined. Drop batter by large spoonfuls on top of apple wedges, spreading evenly to cover apples thinly.
Bake at 350° for 30 to 35 minutes or until cake is golden brown and tester inserted in center comes out clean. Remove skillet from oven, and let stand 3 to 4 minutes. Loosen edges of cake with a small knife. Place a serving plate on top of cake, and invert skillet. Leave skillet over cake 1 minute, then carefully remove skillet. (Replace any apple slices that may have stuck to bottom of skillet.) Let stand about 10 minutes. Serve warm or at room temperature with Caramel Sauce and Butter Pecan Ice Cream.