Bring a saucepan of water to a boil. Cook chicken in boiling water until no longer pink in the center, 7 to 10 minutes; drain.
Bring chicken broth to a boil in a soup pot; add cheese tortellini, reduce heat to medium-low, and cook until tortellini are cooked through and firm to the bite, about 8 minutes. Stir chicken, cream of chicken soup, half-and-half, spinach, thyme, and black pepper with the tortellini; bring to a simmer and continue cooking until hot, 7 to 10 minutes. Serve hot.