1. Place nuts in a 10- by 15-inch baking pan. Roast in a 350° regular or convection oven, shaking pan occasionally, until golden beneath skins, 10 to 12 minutes. Pour nuts into a towel and rub to remove loose skins. Let cool at least 15 minutes. Set aside eight completely skinned nuts. Whirl remaining nuts in a food processor until finely ground.
2. In a bowl, with a mixer on high speed, beat egg yolks and 1/4 cup sugar, scraping bowl occasionally, until very thick and light-colored, about 4 minutes. Stir in ground nuts, bread crumbs, and vanilla.
3. In a large bowl, with clean beaters, beat egg whites on high speed until they hold soft peaks. Gradually add remaining 1/4 cup sugar and continue to beat until egg whites hold short, distinct peaks, about 3 minutes total. Add half the whites to nut mixture and stir to blend well. Gently fold in remaining whites. Spread batter level in a buttered and floured 9-inch cheesecake pan with removable rim.
4. Bake in a 350° regular oven or 325° convection oven until cake is golden brown and springs back in the center when lightly pressed, 25 to 30 minutes. Let cool in pan for 10 minutes. Run a knife between cake and pan rim, then remove rim. Let cake cool on a rack about 45 minutes.
5. With a long, serrated knife, split cake in half horizontally. Gently slide a baking sheet under top cake layer and lift it off. Spread bottom cake layer evenly with chocolate ganache. Slide top layer, cut side down, back in place over ganache.
6. Set cake on rack in a 12- by 17-inch pan. Pour about 1 1/2 cups warm (see notes) caramel cloak over cake--enough to coat it--starting at the center and spiraling to edges, letting caramel drip down sides to cover completely. Arrange reserved hazelnuts evenly around top edge of cake. Let stand until caramel stops dripping and is firm enough to cut, about 30 minutes.
7. Scrape caramel drips from pan back into the measuring cup containing remaining caramel cloak. Cook, uncovered, in a microwave oven at 30% power, stirring occasionally, until warm and fluid, about 2 minutes. Pour into a bowl.
8. Place cake on a plate. Cut into wedges with a sharp knife. Offer remaining caramel cloak to spoon over portions.
Chocolate ganache: In a 2- to 3-quart pan over low heat, stir 8 ounces bittersweet or semisweet chocolate, chopped (about 1 1/2 cups), and 1 cup whipping cream until melted and smoothly blended, 8 to 10 minutes. Let cool, stirring occasionally, until ganache no longer flows when pan is tilted, 2 to 2 1/2 hours.
Caramel cloak: In a 3- to 4-quart pan, combine 1 cup each firmly packed brown sugar, light corn syrup, and whipping cream; 1/2 cup (1/4 lb.) butter; and 1/4 teaspoon salt. Bring to a boil over medium-high heat and stir occasionally until mixture reaches 240°, 12 to 14 minutes. Pour into a 1-quart glass measure and stir occasionally until mixture cools to 150°, about 25 minutes. Stir in 1 teaspoon vanilla. Pour over cake immediately.