Pour 3/4 cup of the broth and 2 cups water into a covered medium-sized saucepan and bring to a boil over high heat. Add the rice, stir, and reduce the heat to low. Cover the pan and simmer until the rice is tender about 20 minutes.
Thaw chicken. Heat oil in an extra-deep 12" skillet over medium heat. Peel and coarsely chop the onion, adding it to the skillet as you chop. Cut the chicken into bite-sized pieces and add them to the skillet as you cut. Sprinkle black pepper over the chicken and cook, stirring often.
While the chicken cooks, add the garlic to the skillet. When the chicken is no longer pink in the center add the cream of chicken soup and the remaining 1 1/4 cups broth to the skillet. Stir until the soup is thoroughly blended and the sauce is smooth. Sprinkle the broccoli pieces evenly over the ingredients. Cover the skillet and cook until heated through.
Sprinkle the cheese evenly over the mixture and reduce the heat to low. Cook covered until the cheese is melted. Serve chicken over rice.