Preheat oven to 350F (175C). Grease a 9-inch-diameter springform pan with 2 3/4-inch-high sides. Line bottom of the pan with parchment paper. Stir coffee liqueur, butter, vanilla and chocolate in a heavy saucepan over low heat until smooth and melted. Cool to lukewarm. Beat eggs and 1 cup sugar in large bowl with an electric mixer until pale and thick, and slowly dissolving ribbon forms when beaters are lifted, for about 6 minutes. Fold 1/3 of the egg mixture into lukewarm chocolate mixture. Fold remaining egg mixture into chocolate mixture.

Place the prepared pan on a baking sheet. Transfer batter to the pan. Bake until a toothpick inserted into center comes out with moist with crumbs attached, or about 55 minutes. Cool for 5 minutes. Gently press down edges of cake. Cool completely in pan. (Cake can be prepared up to 1 day ahead. Cover with plastic wrap and refrigerate. Let stand at room temperature 1 hour before continuing.)

Run a knife around the pan sides to loosen cake. Remove the sides of pan and transfer to a platter. Remove the parchment paper. Dust cake with confectioner's sugar and serve.