Season the chicken with salt and pepper. Whisk together the olive oil, capers, parsley and lemon juice in a bowl. Put the chicken in a resealable bag, pour the marinade into the bag and let marinate for at least 30 minutes and up to 2 hours in the refrigerator.
Prepare a grill for two-level heat. If using a gas grill, set to medium-high heat on one side and low on the other side. If using a charcoal grill, bank hot coals on one side of the grill.
Remove the chicken from the marinade; reserve the marinade. Place the chicken skin-side up on the hot side of the grill, close the lid and grill for 3 minutes. Flip the chicken and grill for 1 minute, then move to the cooler side of the grill skin-side up.
Pour the marinade and stock into a small saucepan and place on the hot side of the grill. Cover the grill and cook until the chicken measures 160 degrees F, about 10 minutes; make sure the marinade is simmering. During the last 2 to 3 minutes of cooking, add the lemon halves cut-side down and cook until charred in spots. Remove the chicken and lemon to a plate.
Whisk the butter into the marinade until combined, then pour over the chicken. Serve with the grilled lemon.
For more ingredient substitution ideas, see the article below.