Preheat the oven to 325°. Line a 9-by-13-inch baking pan with foil, pressing it into the corners and leaving overhang on the 2 short sides.
Spread the hazelnuts in a pie plate and toast for 15 minutes, until fragrant and the skins blister; transfer to a kitchen towel to cool slightly, then rub off the skins. Let cool completely. Transfer the nuts to a food processor. Add the cocoa, salt and 1/2 cup of the sugar and pulse until finely ground.
In a large saucepan, cook the butter over moderate heat, shaking the pan occasionally, until nutty-smelling and golden and the foam subsides, about 5 minutes. Remove from the heat and add the chopped chocolate; let stand until melted, about 2 minutes. Whisk the butter and chocolate until smooth, scraping up any browned butter solids from the bottom of the pan. Let cool slightly.
In a large bowl, using an electric mixer, beat the eggs with the remaining 2 1/2 cups of sugar and the coffee until tripled in volume, about 5 minutes. Beat in the chocolate-butter mixture. Add the cocoa-hazelnut mixture and beat to combine, scraping down the bowl. Scrape the batter into the prepared pan and bake for about 50 minutes, until the top is glossy and a toothpick inserted in the center comes out with a few moist crumbs attached. Let the brownie cool completely on a rack, then refrigerate until chilled.
Lift the brownie out of the pan and peel off the foil. Cut into 20 pieces and serve.