Place all the ingredients in your bread machine, in the order recommended by your bread machine manufacturer. Set your machine on the dough cycle.
Meanwhile make the hazelnut filling: In a food processor, process the sugar and hazelnuts 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 the floured board. Separate the dough into two equal portions. Roll each ball of dough into 4 10 by 8 rectangles. Spread an even layer of filling over all the dough. Then, roll each rectangle, on the long side, up into a rope. Twist the middle of each rope three 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. Brush the rings with beaten egg white.
Bake at 350 degrees for 20 minutes, or until they are 180 degrees inside. Let the cakes cool on a wire rack.
Meanwhile make the icing by beating together all the ingredients. Spread the icing all over the top and sides of the cakes. Make the colored sugar by dividing the sugar between 3 little bags. Add a drop of food color paste to each bag and rub the bag together to distribute the dye. Sprinkle the sugar in stripes all over the tops of the cakes.