Preheat oven to 350°F. Butter 9-inch-diameter springform pan with 2 3/4-inch-high sides. Line bottom of pan with parchment paper. Stir chocolate, butter, coffee liqueur, and vanilla in heavy large saucepan over low heat until melted and smooth. Cool to lukewarm. Using electric mixer, beat eggs and 1 cup sugar in large bowl until thick and pale, and slowly dissolving ribbon forms when beaters are lifted, about 6 minutes. Fold 1/3 of egg mixture into lukewarm chocolate mixture. Fold remaining egg mixture into chocolate mixture.
Place prepared pan on baking sheet. Transfer batter to prepared pan. Bake until tester inserted into center comes out with moist crumbs attached, about 55 minutes. Cool 5 minutes. Gently press down edges of cake. Cool completely in pan.
Run knife around pan sides to loosen cake. Remove sides of pan; transfer cake to platter. Remove parchment paper. Sprinkle cake with powdered sugar and serve with fresh whipped cream or creme faiche