Make the Bread: In the bowl of a stand mixer, whisk together the flour, sugar, yeast, red pepper flakes and salt. In a liquid measuring cup, whisk together the warm water, eggs, egg yolk, and melted butter. Add the egg mixture to the flour mixture in the mixing bowl. Using a dough hook, knead on medium speed until the dough clears the bottom and sides of the bowl, 4 to 8 minutes.
Shape the dough into a ball and transfer to a greased bowl, turning to coat the dough. Cover with plastic wrap and let rise in a warm place until doubled in size, 1½ to 2 hours.
Grease a 9-inch round cake pan; set aside.
Transfer the dough to an unfloured work surface and press to deflate. Roll the dough into an 18x12-inch rectangle with the long side parallel to the counter's edge. Distribute the Monterey Jack and provolone cheeses evenly over the surface of the dough, leaving a 1-inch border around the edges. Starting with the edge closest to you, roll the dough into a log, pinching the ends and seam to seal. Roll the log back and forth, applying gentle pressure, until it reaches 30 inches in length. (If the dough tears in any places, just pinch together to re-seal.)
Starting at one end, wind the log into a coil and tuck the end underneath the coil. Place the loaf in the cake pan and cover loosely with a clean kitchen towel. Allow to rise in a warm place until doubled in size, 1 to 1½ hours.
Adjust oven rack to lower-middle position and preheat oven to 350 degrees F.
Brush the top of the loaf with the beaten egg, then sprinkle with the red pepper flakes. Place the cake pan on a rimmed baking sheet. Bake until the loaf is golden brown, about 25 minutes. Rotate the pan 180 degrees, tent the loaf with aluminum foil, and continue to bake until the loaf registers 190 degrees F, 25 to 30 minutes longer.
Transfer the pan to a wire rack and immediately brush with the softened butter. Let cool for 10 minutes, then run a knife around the edge of the pan and turn the loaf out the wire rack. Cool for 30 minutes before slicing; serve warm or at room temperature. The bread can be kept, wrapped tightly in plastic wrap, at room temperature for up to 4 days.