The goal is to maintain a low enough grill temp so that the chicken cooks low and slow. 4 The chicken is done when you the internal temperature of the chicken pieces are 165°F for breasts and 170°F for thighs, which you test by using a meat thermometer.he goal is to maintain a low enough grill temp so that the chicken cooks low and slow. 4 The chicken is done when you the internal temperature of the chicken pieces are 165°F for breasts and 170°F for thighs, which you test by using a meat thermometer.