Our Caramel Peach Sauce is the key to this delicious layered cake.
Preheat oven to 350 degrees. Trace two 8-inch circles on each of two 12-by-16-inch pieces of parchment paper with a pencil. Place each piece, pencil marks down, on a baking sheet.
In the bowl of a food processor, finely grind almonds and all but 1 tablespoon of sugar, about 1 minute. Transfer to a large bowl.
In the bowl of an electric mixer fitted with the whisk attachment, beat egg whites and salt until soft peaks form, about 1 minute. Add remaining tablespoon of sugar, and beat for 30 seconds.
Fold 2 egg whites into almonds and sugar until well combined. Fold in remaining 4 egg whites until just incorporated.
Divide the mixture evenly among the 4 traced circles. With an icing or rubber spatula, spread mixture to edge of circles. Bake until lightly browned and a cake tester comes out clean, 20 to 25 minutes, switching baking sheets from top to bottom oven shelves twice while baking for even browning. Remove from oven, and transfer cakes, still on the parchment, to wire racks for 3 minutes. Peel off parchment, and return cakes to wire racks to cool completely.
In the bowl of an electric mixer fitted with the whisk attachment, combine mascarpone, heavy cream, confectioners' sugar, and vanilla extract. Beat, beginning at low speed and increasing to medium high, until mixture is thick and fluffy, about 1 1/2 minutes. This cream can be prepared up to 4 hours in advance and refrigerated until you're ready to assemble the cake.
Place one cake layer right side up on a serving plate. Brush lightly with peach schnapps, and spoon a quarter of the mascarpone cream on top. Spread to within 1/2 inch of the edge. Spread a third of the peach puree (reserved when making caramel peach sauce) to cover the cream. Repeat twice, assembling 2 more layers. Place the last cake layer on top, and press down gently so mascarpone cream and peach puree flow to the edges. Brush the top with the remaining peach schnapps, and spread remaining mascarpone cream over the top. Insert 4 or 5 cake testers into the cake, and lay plastic wrap on top to cover cake without touching it. Refrigerate for 1 hour or overnight.
When ready to serve, remove the plastic wrap and cake testers. Lift peach pieces out of the sauce, and arrange on top of the cake. Serve remaining sauce on the side, or spoon over cake slices.
Unblanched almonds produce a more finely textured and lighter-colored cake.