Cut each chicken breast into several strips, then sprinkle with salt and pepper.
Melt butter in heavy large skillet over medium-high heat. Add chicken and cook until brown on both sides, about 5 minutes. Transfer chicken to plate.
Add green and red bell peppers and onion to same skillet and sauté until crisp-tender, about 5 minutes. Add minced garlic and crushed red pepper to skillet and sauté 4 minutes. Add half & half and chicken stock. Simmer until sauce thickens slightly, about 8 minutes.
Cut chicken into strips and add to sauce. Simmer until chicken is cooked through, about 2 minutes. Add basil and 1/4 cup grated Parmesan cheese to sauce, stirring to incorporate. Season sauce to taste with salt and pepper.
Meanwhile, cook pasta in large pot of boiling salted water until tender but still firm to bite. Drain and return to pot. Add sauce and toss to coat.
Serve, passing additional Parmesan separately.