Preheat the oven to 350 degrees F. In a small saucepan, warm the milk and 2 teaspoons of the butter over medium-low heat. With an electric mixer fitted with a wire whip, beat the eggs and 1 cup of the sugar on medium-high speed in a large mixing bowl until the mixture is pale yellow and thick and has tripled in volume, about 8 minutes. With the mixer on low, beat in the warm milk mixture. Sift the flour, baking powder, cocoa, and salt together in a medium-size mixing bowl. Fold into the egg mixture and blend thoroughly until smooth. Add the vanilla and mix gently. Grease a 17 by 12-inch baking pan (or jelly-roll pan) with 2 tablespoons butter. Sprinkle with the remaining 2 tablespoons sugar. Pour the cake batter into the prepared pan, spreading it evenly. Bake until the cake springs back when touched, about 15 minutes. Cool for about 2 minutes, then gently flip it out onto a large wire rack or a large piece of parchment paper. Let cool completely. In an electric mixer, fitted with a wire whip, combine 1 cup of the melted chocolate and cream. Whisk on medium speed until firm peaks form. Remove, cover and refrigerate until ready to use. Place a 1 1/2 quart mixing bowl upside down on the sponge cake. Using a knife, cut out a circle of the cake the size of the bowl, discarding the trimmings. Line the mixing bowl with plastic wrap. Spoon and spread the bottom and sides of the mixing bowl evenly, about 3 inches deep, with the chocolate ice cream. Place the bowl in the freezer and freeze until firm, about 30 minutes. Spread the Peppermint ice cream evenly over the bottom and sides, about 3 inches deep. Spoon the chocolate mousse evenly in the center of the ice cream. Place the circle of sponge cake on top of the filled bowl. Cover the bowl tightly with plastic wrap, place in the freezer, and freeze until firm, about 2 hours. Remove from the freezer and remove the plastic wrap. Unmold onto a wire rack. Pour the remaining 2 cups of melted chocolate over the entire mold, covering completely. Place back in the freezer for about 30 minutes, or until the chocolate is firm to the touch. To serve, using a hot knife, slice the bomb into 12 equal servings. Place each slice on a serving plate and garnish with a drizzle of the chocolate sauce and a sprig of mint.