Grease a 9 inch round cake pan with butter and line the bottom with parchment paper. Grease the parchment paper and remove the sides of the pan.
Mix together the sugars and cinnamon in a bowl.
On a lightly floured surface roll the croissant dough to a 24 x 9 inch rectangle. Trim the edges to make the rectangle straight and brush off excess flour. Sprinkle the cinnamon sugar mixture over top, leaving a one inch border on one of the long sides.
Using a pastry brush, brush the border with the beaten egg, and with the long side facing you, roll the dough into a tight log. Press and stretch the log so it is nicely taut. Seal the seams by pressing them together.
Transfer the log to the base of the pan, connecting the ends to form a circle. Using a sharp knife, cut through the dough halfway, from the top down the middle at two inches interval. Attach the side of the pan and cover the pastry with plastic, let rise for 2 and 1/2 hours.
Preheat oven to 375 F. Brush the surface of the dough with the remaining egg, and then sprinkle the streusel topping. Bake in the center of oven until lightly golden and the streusel is crisp, about 45 minutes.