In the bowl of a stand mixer, add the flour, sugar, yeast, red pepper flakes, and salt. Whisk to combine and fit onto the mixer with the dough hook.
In a medium bowl, whisk together the water, eggs, yolk, and butter.
On medium speed, pour the egg mixture into the flour mixture and let knead until the dough no longer sticks to the side or bottom of the bowl, about 10 minutes.
Remove the dough and form into a smooth ball. Place in a large, greased bowl. Cover loosely with plastic wrap and let sit in a warm place for 1 ½ to 2 hours, until doubled in size.
When ready, turn the dough out onto the work surface and roll into a 12x18-inch rectangle. (If the dough is sticky, generously flour the surface first.) Evenly sprinkle the cheeses over the dough, leaving a 1-inch border untouched. Tightly roll the dough into a log.
Place the log seam-side down and roll it into a spiral. Tuck the end of the spiral under the loaf. Transfer to a greased 9-inch round cake pan or Dutch Oven. Cover loosely with plastic wrap and let sit in a warm place for 1 ½ to 2 hours, until doubled in size.
About 20 minutes before the second rise is through, place oven rack to the lowest position and pre-heat oven to 350°F.
To the risen dough, brush on the egg wash and sprinkle on the red pepper flakes to liking. Bake for 25 minutes. Rotate and tent loosely with aluminum foil. Continue baking for an additional 25 minutes.
Allow to cool in the pan until warm enough to handle. Remove the loaf from the pan and transfer to a cooling rack to cool completely.