Position rack in center of oven and preheat to 350°F.
Butter and flour three 9-inch-diameter cake pans with 1 1/2-inch-high sides. Line bottom of each cake pan with parchment paper round; butter and flour parchment.
Place chopped chocolate in medium metal bowl. Set bowl over saucepan of barely simmering water and stir until chocolate is melted and smooth. Remove bowl from over water and set aside.
Whisk flour, baking powder, baking soda, and salt in medium bowl to blend.
Using electric mixer, beat butter and 1-1/4 cups sugar in large bowl until fluffy and pale yellow, about 2 minutes. Add egg yolks 1 at a time, beating until well blended after each addition. Beat in lukewarm melted chocolate, then stout and coffee. Beat flour mixture into chocolate mixture in 2 additions just until incorporated.
Using clean dry beaters, beat egg whites and remaining 3 tablespoons sugar in another medium bowl until stiff but not dry. Fold 1/3 of egg whites into cake batter to lighten; then fold in remaining egg whites in 2 additions. Divide batter between prepared cake pans (about 2 cups for each); smooth tops.
Bake cakes until tester inserted into centers comes out clean, about 30 minutes.
Transfer cakes to racks and cool in pans 20 minutes. Invert cakes onto racks; remove parchment paper and cool completely.
DO AHEAD: Can be made 1 day ahead to this point. Cover and store at room temperature.
Using a serrated knife, trim cake layers so that tops are flat.
Place 1 cake layer, trimmed side up, on cake plate. Spread frosting evenly to edges of layer. Top with second cake layer, trimmed side up. Spread frosting evenly to edges of layer. Top with third cake layer, trimmed side down. Spread remaining frosting evenly over top and sides of cake.
DO AHEAD: Can be made up to 1 day ahead. Cover with cake dome and refrigerate. Let cake stand at room temperature at least 1 hour and up to 3 hours before serving.