Preheat the oven to 350°. Butter and flour a 9-inch round cake pan. Toast the hazelnuts on a rimmed baking sheet, stirring occasionally, for about 10 minutes, or until lightly browned. Let the hazelnuts cool on the baking sheets, then transfer them to a kitchen towel and rub them together to remove the skins. Transfer the hazelnuts to a food processor and finely chop.
In a small bowl, sift the flour with the baking powder and salt. In a large bowl, using a handheld mixer, beat the butter with the granulated sugar until fluffy. Beat in the eggs, 1 at a time, until incorporated. Using a rubber spatula, fold in the flour mixture in 3 batches, alternating with the rum and hazelnuts.
In a medium stainless steel bowl, beat the egg whites until stiff. Beat one-third of the egg whites into the batter, then gently fold in the remaining whites. Scrape the batter into the prepared pan and bake for about 35 minutes, or until the cake springs back when lightly pressed. Transfer to a rack and let cool in the pan for 15 minutes. Turn the cake out onto the rack and cover loosely with a kitchen towel until cooled to room temperature. Turn the cake right side up and dust the top with confectioners' sugar, then cut into wedges and serve.
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