For the chicken
1/4 cup packed dark brown sugar
1/4 cup soy sauce
2 tablespoons sake or dry sherry
4 medium garlic cloves, finely chopped
3 pounds boneless, skinless chicken breasts or bone-in, skin-on chicken pieces
For the sauce
1/2 cup packed dark brown sugar
1/2 cup soy sauce
1/4 cup naturally sweetened pineapple juice
1/4 cup sake or dry sherry
2 tablespoons white wine vinegar
2 medium garlic cloves, finely chopped
2 tablespoons cornstarch dissolved in 2 tablespoons cold water
To grill and serve:
Vegetable oil, for rubbing the grill grates
2 teaspoons toasted sesame seeds, for garnish
For the chicken:
Place the brown sugar, soy sauce, sake or sherry, and garlic in a large resealable plastic bag and stir to combine. Add the chicken, seal the bag (pressing out any excess air), and turn to coat the chicken evenly. Place in the refrigerator and let marinate, turning occasionally, for at least 2 hours or up to 24 hours.
For the sauce:
Place all of the ingredients except the cornstarch-water mixture in a small saucepan and stir to combine. Bring to a boil over medium-high heat. Reduce the heat to low and simmer until the sauce has reduced to about 1 1/4 cups, about 15 minutes.
Add the cornstarch mixture, whisk to combine, and cook until the sauce has thickened, about 1 minute. Divide the sauce equally between 2 small bowls; set aside 1 bowl for serving and the other for brushing onto the chicken while it cooks on the grill.
To grill and serve:
When ready to grill, remove the chicken from the refrigerator and let it sit at room temperature for about 15 minutes. Meanwhile, heat a gas or charcoal grill to medium (about 350°F to 450°F).
When the grill is ready, rub the grates with a towel dipped in vegetable oil. Remove the chicken from the bag and place it on the grill (skin-side down if using skin-on), cover the grill, and cook until grill marks appear on the bottom, about 5 minutes. Flip the chicken pieces, cover the grill, and cook until grill marks appear on the second side, about 5 minutes more.
Flip the chicken again and generously brush the pieces using the bowl of sauce set aside for that purpose. Cover the grill and cook for 5 minutes. Flip and brush the chicken, and continue to flip and brush every 5 minutes, until an instant-read thermometer inserted into the thickest piece reads 160°F to 165°F and the meat near the bone is no longer pink, about 10 to 15 minutes more (about 25 to 30 minutes total cooking time). Remove to a clean serving platter, sprinkle with the toasted sesame seeds, and serve, passing the remaining bowl of barbecue sauce on the side.