In 4-quart saucepan, melt butter over medium heat. Cook mushrooms in butter 3 to 4 minutes or until softened. Add rice; cook and stir 2 minutes.
Add broth; heat to boiling. Boil 2 minutes. Stir in cooking sauce and thyme; heat to boiling. Reduce heat to medium-low; cover and cook 15 minutes, stirring twice. Remove from heat; let stand about 5 minutes or until rice is tender. Sprinkle with cheese and pepper.