Heat large saucepan of salted water to boiling. Heat butter and oil in large heavy skillet over medium heat. When foam subsides, add chicken, onions, mushrooms and garlic; cook, stirring occasionally, until chicken begins to brown, about 7 minutes.
Stir in cream, tomatoes, peas, salt, and pepper; heat to boiling over high heat. Reduce heat to medium and simmer until sauce is slightly thickened and chicken is cooked through, 3-5 minutes. Stir in basil and remove from heat.
Meanwhile, cook pasta in boiling water until tender, but still firm to the bite. Drain thoroughly and transfer to wide serving bowl. Pour sauce over pasta; sprinkle with Parmesan and toss to combine. Serve hot.