Bring 4-1/4 cups water to a boil; add pilaf mixes, 1 of the flavoring packets (discard second packet) and white rice. Return to boil, then reduce heat to low. Cover and cook 25 to 30 minutes.
Meanwhile, heat large nonstick skillet over medium-high heat; add oil, then butter. Season chicken with 1/2 teaspoon of the salt, 1/4 teaspoon of the pepper and 1/2 teaspoon of the thyme. Add chicken in a single layer and cook 2 to 3 minutes per side; remove to plate and cover. Repeat with remaining chicken.
Add mushrooms and remaining salt, pepper and thyme. Cook, stirring, 2 to 3 minutes or until tender. Raise heat to high; add marsala and cook, stirring, for 2 minutes; add tomatoes and 1 cup of the broth; cook 5 more minutes. Stir remaining 1/2 cup broth and cornstarch together. Reduce heat to medium-low; stir cornstarch mixture into pan. Add chicken and any liquid. Cover and simmer another 5 minutes until chicken is heated through.
Place rice in a serving bowl. Place chicken on platter and pour sauce on top.