Step 1Place an 8-inch skillet on low heat and add the butter. Cut the vanilla bean in half lengthwise. Scrape the seeds out; add the seeds and bean to the butter. When the butter turns a golden brown, 15 to 20 minutes, remove from the heat. Remove the seeds and pod and scrape the butter into a bowl to cool.
Step 2Heat the oven to 350 degrees. Spray a 10-inch cake pan with nonstick spray, and line it with parchment paper. Grind the hazelnuts with the powdered sugar in a food processor.
Step 3Place the ground nuts in a large mixing bowl. Add the flour and stir until well combined.
Step 4Beat the egg whites and granulated sugar to stiff peaks using an electric mixer. Fold the whites into the nut mixture. Fold in the room-temperature brown butter.
Step 5Pour the batter into the prepared pan. Bake until a skewer inserted in the center comes out clean, about 1 hour. Cool for 10 minutes, then run a sharp knife around the edge and unmold onto a serving platter. Sprinkle each serving of cake with a little fleur de sel. Accompany with a scoop of caramel ice cream.