Chocolate Torte with Calvados-Poached Figs

Place a rack in middle of oven; preheat to 375°F. Butter a 9"-diameter springform pan or cake pan and line with a round of parchment. Butter parchment and sprinkle pan with superfine sugar.
Melt chocolate in a heatproof bowl set over a saucepan of simmering water (bowl should not touch water), stirring, until smooth; let cool slightly.
Pulse hazelnuts, flour, cocoa powder, salt, and 1/2 cup superfine sugar in a food processor until finely ground; sift through a medium-mesh sieve. Pulse any larger pieces again in food processor, if needed.
Using an electric mixer on high speed, beat 1 cup butter and remaining 3/4 cup superfine sugar until light and fluffy, about 4 minutes. Add egg yolks, 1 at a time, beating to blend between additions. Beat in vanilla. Reduce speed to low; mix in melted chocolate. Gradually add dry ingredients, mixing just until combined.
Using clean beaters and with mixer on medium-high speed, beat egg whites and cream of tartar in a medium bowl until stiff peaks form.
Using a rubber spatula, fold 1/3 of whites into chocolate batter to loosen, then fold remaining whites into batter just until combined. Scrape batter into prepared pan.
Bake cake 20 minutes, then reduce oven temperature to 350°F and bake until a tester inserted into the center comes out clean, 30–40 minutes longer. Transfer pan to a wire rack; let cake cool 30 minutes before unmolding to cool completely.
DO AHEAD: Cake can be baked 2 days ahead. Store tightly wrapped at room temperature.
Bring Calvados, granulated sugar, and 2/3 cup water to a boil in a small saucepan, stirring to dissolve sugar. Add figs, reduce heat to low, and simmer gently until figs are very soft, 20–25 minutes. Let cool.
Serve cake with figs and syrup.
DO AHEAD: Figs can be poached 1 week ahead. Cover and chill.