Preheat the oven to 325°F. Grease a 9-inch springform pan with 1 Tbs. of the butter. Place the pan on top of a sheet of parchment and trace a circle around the pan, cut it out, and place the paper circle in the bottom of the pan to grease the underside. Flip the paper circle over and press it into place. Set aside.
Place the bread in a food processor and pulse until it forms medium-coarse crumbs. Add the peanuts and peanut flour, and pulse until the texture is like rough sand (don’t overprocess the mixture or it will become peanut butter).
Use a standing mixer fitted with the paddle attachment or a large bowl (if using a hand mixer) and cream the remaining 10 Tbs. butter until it is fluffy and pale. Add the brown sugar and cream until pale, then add the egg yolks, one at a time, mixing thoroughly between additions and scraping the bottom and sides of the bowl as needed. Add the vanilla bean paste (or extract), reduce the speed to low, and add the peanut mixture.
In another bowl, whisk the egg whites with 3 Tbs. of the sugar until the whites form stiff peaks. Fold the whipped whites into the batter in 3 additions, adding the grated chocolate along with the last addition, folding until just a few streaks of white remain. Scrape the mixture into the prepared cake pan and bake until the sides pull away from the edges of the pan and a cake tester inserted into the center of the cake comes out clean, about 1 hour.
Remove the pan from the oven and set on a wire cooling rack for 5 minutes before releasing the latch and lifting the side of the pan away from the bottom. Cool completely before inverting the cake onto a large flat plate, cutting board, or baking sheet, lifting off the pan bottom, and peeling away the parchment paper circle. Reinvert the cake onto a cake plate or platter.
Place the remaining 1/2 cup sugar and 2/3 cup of cold water into a small saucepan. Cook over medium heat, stirring until the sugar dissolves, and then let the sugar simmer, swirling the pan occasionally, until the liquid becomes an amber-color caramel, about 15 to 17 minutes. Remove from the heat immediately and, using an offset spatula, pour and spread the caramel syrup over the top of the cake. Cool the cake and then slice and serve at room temperature.