Heat broth in medium saucepan, and keep at a simmer over low heat. In a large saucepan, combine 2 TBS butter with onion and olive oil. Cook over moderate heat, stirring occasionally, until onion is light golden brown. Add tomatoes, and a small amount of salt and pepper. Cook, stirring occasionally, for 10 minutes.
Add rice and stir to coat grains. Add 1/2 cup of broth to rice and stir constantly until most of liquid has been absorbed. Gradually add more broth, 1/2 cup at a time, and continue stirring. The rice is done when firm, but tender, with no chalky center.
Add remaining 2 TBS of butter, cheese, and basil. Stir for 2 minutes. Season to taste.