reheat your oven to 375 and line a 10 by 15 inch jelly roll pan with aluminum foil. Spray the foil with cooking spray and sprinkle with flour.
Sift the flour, baking powder, sugar, and cream of tartar together, mix until just combined.
In a stand mixer with a whisk attachment, beat the eggs and 1/3 cup of water together on high until they are light and airy, about five minutes.
Gently fold the egg mixture into the flour mixture, being careful not to over mix.
Divide the batter into two, and dye one half red.
Running parallel to the 10 inch side of the pan, gently pour a spoonful of the red cake batter all along one edge. Then a row of white, then a row of red, and so on. I had four rows of each color in the end, and had to work once all the batter was added to the pan to make sure it was completely spread out.
Bake 10 to 12 minutes or until the white begins to turn golden brown.
While the cake is baking, spread a clean kitchen towel on a large cutting board. Use your 1/4 cup of powdered sugar to cover the kitchen towel completely.
When the cake is done baking, flip the entire pan over onto the kitchen towel and gently peel away the aluminum foil.
Working at one end, roll the cake into the kitchen towel. This will give the cake its rolled form. Leave the cake for at least one hour, but not more than four hours.
While you are waiting to unroll your cake, mix together two sticks of softened butter, two cups of powdered sugar, and the peppermint extract. Don't refrigerate the frosting while you are waiting for your cake!
Uroll your cake. Spread the frosting evenly over the cake and roll it back up. (This time without the towel ;)
Roll the cake in aluminum foil and chill for at least an hour before serving, but do not wait more than a day or two to serve or the cake will dry out.
When you are ready to serve, whisk together the topping of powdered sugar and milk. Pour over cake (you may need to do some spreading) and top with crushed candy canes.