Prepare a hot fire in a charcoal grill or heat a gas grill or broiler. Away from the heat source, lightly coat the grill rack or broiler pan with cooking spray. Position the cooking rack 4 to 6 inches from the heat source. Grill or broil the chicken until browned and just cooked through, about 5 minutes each side. Transfer the chicken to a cutting board and let rest 5 minutes before slicing into strips.
In a large, nonstick frying pan, heat the olive oil over medium heat. Add the onions and mushrooms and saute until tender, about 5 minutes. Stir in the garlic, basil and grilled chicken strips. Keep warm.
Fill a large pot 3/4 full with water and bring to a boil. Add the pasta and cook until al dente (tender), 10 to 12 minutes, or according to the package directions. Drain the pasta thoroughly. Return the pasta to the pot and add the chicken mixture. Toss to mix evenly.
Divide the pasta among the plates. Garnish each with 1 tablespoon Parmesan cheese and black pepper. Serve immediately.