In a medium mixing bowl beat the 1/3 cup sugar, margarine or butter, shortening, and 1/4 teaspoon vanilla on medium speed of an electric mixer until combined. Blend in flour and salt until crumbly. Pat onto the bottom of a 9-inch springform pan. Set aside.
Place apple slices in a single layer in a shallow baking pan. Cover with foil. Bake in a 400 degree F oven for 15 minutes.
Meanwhile, for filling, in a large mixing bowl beat the cream cheese, the 1/2 cup sugar, and the 1/2 teaspoon vanilla with an electric mixer until fluffy. Add the eggs, beating on low speed just until combined. Pour into dough-lined pan. Arrange warm apple slices atop filling. Combine the 1/3 cup sugar and the cinnamon. Sprinkle filling with sugar mixture and the almonds. Bake in a 400 degree F oven for 40 minutes or until golden and filling is set; cool. Remove sides and place cheesecake on a serving plate. Cover; chill 4 to 24 hours before serving. Makes 12 servings.