Preheat the oven to 350°. Lightly spray or wipe a 10-1/2-inch springform pan with vegetable oil.
Combine the chestnuts, 1/2 cup of the sugar, the cognac, and the vanilla seeds in a medium saucepan over high heat. Bring to a boil; continue to boil for 5 to 8 minutes, stirring often, until very thick. Cool the mixture and remove the vanilla seeds. Lightly crush the chestnuts.
Place the ricotta, the remaining sugar, and the cottage cheese in a blender; process until smooth. Add the eggs, egg whites, and vanilla extract. Pulse just until combined; do not overmix. Transfer the mixture to a mixing bowl.
Lace the chestnut pieces through the cheese mixture, creating streaks like a marble cake. Pour the mixture into the springform pan, smoothing the top. Bake for 1 hour; reduce the oven temperature to 325°, and bake for another 15 minutes, or until a knife inserted in the cake 2 inches from the edge comes out clean.
Serve at room temperature, or chill until ready to use. Remove from the pan, slice, and serve.
SERVING SUGGESTION: top with Vanilla Chestnut Sauce.