Preheat the oven to 400 degrees F. Combine the mozzarella, feta and butter in a large bowl. Slice off the top 2 inches of the bread boule and hollow it out, reserving the top and insides for dipping. Fill the bread bowl with the cheese mixture. Transfer to a baking dish, cover loosely with foil and bake 20 minutes.
Uncover and crack the egg on top of the cheese mixture. Continue baking, uncovered, until the egg white is almost set, 12 to 15 minutes, then broil until just set, 1 to 2 more minutes.
Remove from the oven and sprinkle with the red pepper flakes. Just before serving, break the egg yolk with a fork and beat into the melted cheese.