1. In a 12-inch frying pan or 5- to 6-quart pan over high heat, melt 1 1/2 cups butter and stir often until the milk-solid particles turn golden brown, 5 to 7 minutes. Remove at once from heat and let stand until warm, 15 to 20 minutes (or nest pan in cold water and stir often, 5 to 8 minutes).
2. Meanwhile, in a food processor, whirl 3/4 cup slivered almonds to a fine powder (or finely grate with a nut grater); scrape into a large bowl. Spread remaining 1 cup slivered almonds in a 10- by 15-inch pan.
3. Bake in a 350° oven until slivered almonds are golden brown, 6 to 12 minutes. Let cool slightly, then coarsely chop.
4. To bowl with ground almonds, add toasted almonds, sugar, and 1 1/4 cups flour. Stir to mix well. Scrape melted butter with all browned particles into bowl; mix. Add egg whites, lemon peel, and orange peel. Beat with a mixer on medium speed, scraping bowl often, until batter is well blended. Add chopped apples and stir to mix.
5. Butter and flour a 9-inch cheesecake pan with removable rim. Scrape batter into pan.
6. Bake in a 325° oven (300° convection oven) until cake is golden brown, firm in the center when lightly pressed, and begins to pull from pan sides, 1 hour to 1 hour and 10 minutes. Let cake stand in pan on a rack at least 20 minutes or until cool, about 1 hour. Slide a thin-bladed knife between warm or cool cake and pan rim; remove rim. Set cake on a rimmed plate and, if desired, pour about 1/2 cup warm caramel sauce over the top. Cut into wedges; serve with scoops of ice cream and remaining caramel sauce to add to taste.