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Cut each chicken breast into several strips, then sprinkle with salt and pepper.
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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.
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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.
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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.
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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.
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Serve, passing additional Parmesan separately.