In a bowl, mix together the shredded chicken, buffalo sauce, softened cream cheese, ranch or blue cheese dressing, mozzarella, and cheddar until well combined.
Preheat your oven to 375°F (190°C). Lightly flour a surface and roll out the pizza dough into a large rectangle, about 10x14 inches.
Spread the buffalo chicken mixture evenly over the dough, leaving about 1 inch of space around the edges.
Starting from the long side, tightly roll the dough into a log (like cinnamon rolls), pinching the seam to seal. Tuck the ends underneath.
Place the roll seam-side down on a parchment-lined baking sheet. Brush the top with the beaten egg. Bake for 30–35 minutes, or until golden brown and the center is cooked through.
Let the bread rest for 5–10 minutes before slicing into thick pieces. Serve with extra buffalo sauce, ranch, or blue cheese dressing for dipping.