Cook tortellini according to package directions. Drain but reserve water.
While the pasta cooks, melt butter in a large skillet over medium-high heat. Add onions, mushrooms, and garlic, cooking until softened, about 4 minutes.
Stir in artichokes and chicken. Toss together with cooked tortellini.
Gently mix in spinach, roasted red pepper, and cheeses. Heat through and serve, or if you prefer a creamier sauce, add one cup of pasta water and simmer for about two minutes.