Preheat the oven to 375°F (190°C) and line a baking sheet with parchment paper.
Heat a small skillet over medium-high heat and melt 1 tablespoon unsalted butter.
Add the thinly sliced red onion and cook, stirring, until tender and slightly softened, about 3–4 minutes; remove from heat and set aside.
On a piece of parchment, roll out the tube of thin-crust refrigerated pizza dough into a rectangle.
In the center of the dough, arrange 6 slices provolone in a row, then top with 2 cups shredded chicken, the cooked onions, and 1/2 cup barbecue sauce.
Fold the longer sides over the filling one at a time, then fold the shorter ends up; trim any excess dough and pinch seams to seal the stromboli.
Carefully flip the stromboli so the seam side is down on the parchment-lined baking sheet.
Brush the outside with the beaten egg, then sprinkle with 1 teaspoon Italian seasoning and 1 teaspoon grated Parmesan cheese.
Bake for 20–25 minutes, or until the crust is golden brown on top; remove from oven and let rest a few minutes before slicing.
Serve slices with extra barbecue sauce on the side for dipping, if desired.