Preheat your oven to 425°F (220°C). Let it fully heat so the dough bakes up crisp instead of dense. Lightly grease a baking sheet with olive oil or line it with parchment paper.
In a large bowl, toss the 2 cups cooked chicken with ¼ cup buffalo wing sauce. Stir in the ¼ cup ranch dressing and remaining ¼ cup buffalo sauce until evenly coated and creamy. The mixture should be glossy and slightly thick, not watery. If it’s too loose, mix in a small handful of shredded cheese to tighten it up.
Lightly flour your countertop and roll the defrosted pizza dough into a long rectangle, about 10x14 inches. If it springs back, let it rest for 5–10 minutes, then roll again.
Spread the buffalo mixture evenly over the dough, leaving a 1-inch border around the edges. Sprinkle mozzarella first, then cheddar, distributing evenly. Don’t overfill, or it may split in the oven.
Starting from one long side, roll the dough into a tight log. Pinch the seam closed and seal both ends well. Place seam-side down on the baking sheet and brush the top lightly with 1 tsp olive oil.
Bake for 30 minutes at 425°F (220°C), until deep golden brown and hollow-sounding when tapped. If it browns too quickly, tent loosely with foil for the last 5–10 minutes. The internal temperature should reach at least 165°F for safe serving.
Let it rest 5–10 minutes before slicing so the cheese can set. Slice thick and serve warm with pizza sauce, ranch, or blue cheese dressing.