Heat oil in a large skillet over medium-high heat. Place chicken on a plate and season with thyme, salt and pepper. Add garlic to skillet and cook, stirring, 1 minute. Add chicken to skillet and 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.
In a cup, dissolve cornstarch in remaining 1/2 cup of broth; add to skillet.
Simmer until mixture thickens, stirring constantly, about 1 minute; serve over rice. Yields about 1 cup of chicken and broccoli and 1/2 cup of rice per serving.