Add remaining 1 tablespoon olive oil to saucepan; stir in rice and onion. Cook for 2 minutes. Slowly stir in 1 cup of hot broth mixture. Reduce heat to medium low. Cook, stirring frequently, until all liquid is absorbed. Add remaining broth mixture 1 cup at a time, stirring constantly until the broth has been absorbed. Rice should be creamy and al dente.
Stir in tomatoes and chicken. Heat through. Stir in Asiago cheese, and basil. Serve immediately.