Cut a few deep slashes in the meaty part of each chicken leg. Place the chicken in a large resealable plastic bag and pour in the marinade. Press the air out of the bag and seal tightly. Turn the bag to distribute the marinade, place in a bowl, and refrigerate for 4 to 6 hours, turning the bag occasionally.
Prepare the grill for direct and indirect cooking over medium-high heat (400° to 500°F).
Remove the chicken from the bag and discard the marinade. Place the chicken, skin side down, over indirect medium-high heat. Place a sheet pan on top of the chicken and weight it down with two foil-wrapped bricks. Close the lid and cook until golden around the edges, 20 to 25 minutes. Remove the bricks and sheet pan and, using a large spatula, carefully turn the chicken legs over, being careful not to tear the skin. Replace the weight, close the lid, and cook until the juices run clear and the meat is no longer pink at the bone, about 20 minutes more. If desired, to crisp the skin, remove the weight, turn the chicken skin side down, and grill over direct medium-high heat for about 5 minutes. Remove from the grill, garnish with the cilantro and mint, and serve warm.