Place the rack in the middle of the oven and preheat it to 375°F Spritz a 9x13 shiny aluminum baking pan with Pam or Baker's Joy. Place the unbroken eggs in a bowl of hot tap water.
Beat the butter and sugar in the stand mixer until pale and fluffy, about 3 minutes. Add the eggs, one at a time, then the vanilla, and beat until thoroughly mixed, about 5 minutes. Sift the flour, baking powder, and salt onto a sheet of wax paper. Spoon in the flour in thirds, alternately with milk. Mix just until the batter is smooth.
Pour the batter in the prepared pan and bake until the cake begins to pull away from the sides and a wooden pick comes out clean, 18 to 20 minutes. Cool on a rack.
While the cake is cooling, place the butter in the mixer bowl and cream until soft, adding the egg yolk, salt and vanilla. With the mixer on low speed, add the confectioners' sugar and milk (or cream) alternatively until you have a smooth mixture. Refrigerate until the cake is cool.
Assemble the cake. Invert the cake onto a large serving platter. Smooth the frosting over the top and sides, using an offset spatula. Make a design on top, using the berries to create a flag. Use both raspberries and blueberries in the upper left quadrant to represent the "starts", then make the "bars" by making alternating horizontal stripes of raspberries and blueberries. Finish by alternating whole blueberries and raspberries all around the base of the cake.