Bring 1 1/3 cups broth to a boil in a medium saucepan; add rice to pan. Cover, reduce heat, and simmer 5 minutes or until liquid is absorbed. Set aside.
Heat oil in a Dutch oven over medium-high heat. Add onion and next 4 ingredients (through thyme) to pan; sauté 3 minutes, stirring occasionally. Stir in butter and mushrooms; sauté 3 minutes or until lightly browned. Add remaining 2 2/3 cups broth, rice, chicken, salt, and pepper to pan; cook 3 minutes or until thoroughly heated, stirring occasionally.