While it warms up, lay the flatbreads on a large sheet pan and spread the BBQ sauce over both, then evenly scatter the mozzarella, red onion, mushrooms, chicken, and banana peppers across the two.
If you like extra sauciness, add a bit more BBQ sauce on top before baking.
Slide the pan into the oven and bake for 8-10 minutes, until the cheese has melted and everything is heated through. Finish with a scattering of arugula, slice into pieces, and serve right away.
Notes
Use already-cooked chicken, like rotisserie, since the short bake time won't cook raw chicken through.
Spread the BBQ sauce in a thin layer to keep the flatbread from getting soggy.
Add the arugula after baking so it stays fresh and peppery instead of wilting.
Watch the oven starting around 7 minutes; thin flatbread edges can brown fast.
Leftovers keep 2 days; reheat in the oven, not the microwave, to keep the base crisp.
Toppings can be prepped a few hours ahead and assembled right before baking.