Heat oil in a large skillet over medium-high heat. Add garlic and cook 1 minute. Season chicken with thyme, salt and pepper and add to skillet. Cook until browned on all sides, stirring frequently, about 3 minutes.
Add broccoli to skillet, cover and cook 2 minutes. Add 1 cup of broth, cover and simmer until chicken is cooked through, about 5 minutes.
Dissolve cornstarch in remaining 1/2 cup of broth and add mixture to skillet. Simmer until mixture thickens, stirring constantly, about 1 minute. Serve over rice. Yields about 1 cup of chicken mixture and 1/2 cup of rice per serving.