Place all the dough ingredients in your bread machine, in the order recommended by your bread machine instructions. Set your machine on the dough cycle.
While the dough is in the bread machine, make the Pistachio Filling: In the food processor process the sugar and pistachios into a paste, and then add the rest of the ingredients and process until combined.
When the cycle is complete, empty the dough on to a floured board. Separate the dough into four equal portions. Roll each portion into an 8x10 rectangle. Spread a quarter of the filling on to each rectangle. Roll each rectangle, on the ten inch side, up into a rope. Twist the middle of each rope a few times. Then, take a rolling pin and slightly flatten each rope into an even thickness.
Divide the ropes between two parchment lined baking sheets. Bring the ropes together to form two rings, and gently press them together to seal them. Cover loosely with plastic wrap and let them rise in a warm place for one hour. Preheat the oven to 350 degrees.
Brush each cake with egg whites and bake for 20 minutes or until they are 180 degrees in the middle. Let the cakes cool on a wire rack.
Make the icing by beating together all the ingredients. Add more cream if you need to. It should be a spreadable consistency. Cover the cake top and sides in the frosting.
To make the Colored Sugar: Divide the sugar between three plastic bags. Add a drop of food color paste to each bag and rub the bag together to distribute the dye. Sprinkle it in stripes over the cakes.