Preheat oven to 400 degrees F. Remove the giblets and any fat from the chicken cavity. Roll the lemons on the counter with your hand to soften, then prick with a fork, going all the way through the rind to the flesh. Cut one of the lemons in half.
In a small bowl, mash the garlic with the salt until a paste forms. Rub half this paste inside the chicken and then stuff one and one half of the lemons into the cavity.
Add the butter to the rest of the garlic paste and rub the mixture on the outside of the chicken. Place in a shallow roasting pan, on a rack (you can use sliced onions for a rack for even more flavor) and pour the broth into the bottom of the pan. Squeeze the juice from the lemon half into the broth.
Roast for 60-65 minutes, basting with the pan juices halfway through the cooking time, until a meat thermometer inserted into the thickest part of the thigh registers 180 degrees F, juices run clear when pricked with a fork, and the drumstick moves easily in its socket. Cover the chicken and let rest for 10 minutes before carving. Serve with the pan juices.