Put the thyme, onion, carrot,and chicken breasts in a medium saucepan. Cover with the broth, and bring to a boil. Lower the heat to very low and cover. Poach the chicken for 20 minutes or until firm to the touch. Remove the pan from the heat, uncover, cool the chicken in the liquid for 30 minutes.
Transfer the chicken to a cutting board and reserve the liquid. Bone and skin the chicken and cut the meat into cubes. Discard the bones and skin.
Strain the broth and store, covered, in the refrigerator for 3 days or freeze for later use. Remove any fat from the surface of the broth before using.