Preheat the oven to 425°F (220°C).
Toss the cooked chicken with 1/4 cup of buffalo wing sauce. Adjust sauce quantity based on desired heat level. Stir in ranch dressing and the remaining 1/4 cup of wing sauce until evenly combined.
On a floured surface, roll out the pizza dough into a long rectangular shape.
Spread the buffalo ranch sauce mixture evenly across the dough. Add the chicken, then top with the shredded mozzarella and cheddar cheeses.
Roll the dough from one side to form a log. Seal both ends to prevent filling from oozing during baking, and place the log seam-side down on a non-stick or greased baking sheet.
Bake in the preheated oven for 30 minutes. Allow the stromboli to rest for 5-10 minutes before slicing.
Slice into portions and serve with pizza sauce, ranch, or blue cheese dressing for dipping if desired.