Preheat the oven to 375 degrees F.
Place the sugar, whole eggs, egg yolks, salt, corn syrup, water and vanilla in a heat resistant medium-sized mixing bowl. Set it over a saucepan of simmering water and beat the eggs using a hand held mixer until the eggs have tripled in volume and have reached a temperature of 110 degrees F, about 10 to 15 minutes.
In a separate bowl, sift together the cake flour and baking powder. Gently fold the flour into the beaten eggs in 3 or 4 stages, folding until just combined and being careful not to overmix the batter. Lightly grease an 11 by 17-inch sheet pan with melted butter and line with parchment paper. Butter the top of the parchment paper. Pour the batter into the sheet pan and bake until the center is set and springs back when pressed lightly, about 15 minutes. Remove the cake from the oven and allow it to cool completely before proceeding.
Once the cake has cooled, remove from the pan and lay topside up on a cutting board. Insert toothpicks into the 4 sides of the cake, positioned about 1-inch apart and inserted horizontally about 1/2-inch from the surface of the cutting board. These should serve as a guide for cutting the top brown edge from the cake. Using a long-bladed, serrated knife, cut the top brown edge off of the cake by running the knife against the edges of the toothpicks. Rotate the knife as you cut to make a smooth cut. Carefully turn the cake over and repeat, so that both the top and bottom browned edges have been removed and the cake is about 1/2-inch thick with no visible brown top or bottom. (If it helps, you can use a thin book as a guide and lay the toothpicks on the book, then insert into the cake.)
With the cake trimmed, lay it on a cutting board with 1 of the short ends facing you. Remove the ice cream from the freezer. If the ice cream is in a paper container, use a serrated knife to cut the ice cream in its container into 1/2-inch rings. Peel the paper away, and lay the ice cream on the cake, leaving a 1-inch border on 3 of the sides and a larger border on the short edge that is furthest from you. There should be some ice cream left over; use this to fill in any gaps on the cake that are not covered with ice cream. Starting with the end closest to you, roll the cake into a log, and wrap it in parchment paper. Wrap again in plastic wrap. Place the wrapped cake in the freezer and let freeze at least 2 hours, and up to 1 day before serving.
Once the cake has chilled sufficiently, remove it from the freezer and unwrap it. Trim about 1/2-inch from both edges of the cake using a serrated knife. Slice the cake into 10 or 12 even rings. Place the cocoa powder in a small fine-meshed sieve and sprinkle around the edges of the plate. Run your finger around the edge of the plate to form a clean edge. Using a birthday candle or other object with a fine tip, write your greeting or a guest's name, etc., in the cocoa powder along the plate's edge. Place the cake slice onto the center of the plate and serve with the Raspberry Coulis either drizzled over the cake or on the side. Garnish with the toasted almonds and mint leaves.
Yield: 10 to 12 servings