Marinate the chicken: Place the chicken in a large resealable gallon-sized plastic bag and add half of the can of cola, 2 sliced garlic cloves, 2 smashed ginger coin and 1/2 cup soy sauce. Seal the bag, put it in a rimmed dish and marinate for at least 2 hours or up to overnight in the refrigerator, flipping the bag periodically for even distribution of the marinade.
Make the glaze: Meanwhile, combine the sake with 3 tablespoons brown sugar, 1/3 cup water and the remaining 1/2 cup soy sauce, 4 sliced garlic cloves, and 2 smashed ginger coins in a small saucepan. Bring to a gentle simmer and cook for 5 minutes to develop flavor. Using a fork, stir the cornstarch in a small bowl with 1 tablespoon water. Whisk the cornstarch slurry into the sauce. Simmer the sauce for 1 minute more. Remove from the heat and add 1 teaspoon vinegar. Cool to room temperature, then strain, discarding the solids. Set aside 1/4 cup for drizzling at the end.
Make the pickled peppers: Place the sliced peppers in a medium bowl. Bring 1/2 cup of water to a boil with 1/2 teaspoon salt, then pour over the peppers. Let sit 10 minutes, then drain. Return the peppers to the same bowl. Add 1/4 cup vinegar, 2 tablespoons water, 1/2 teaspoon salt and remaining tablespoon of brown sugar, and stir to dissolve. Set aside.
Take the chicken out of the fridge 30 minutes before cooking. Remove from the marinade, and pat dry with paper towels. (Discard the marinade.)
Prepare a grill for medium indirect heat: For gas grills (with 3 or more burners), turn all the burners to medium-high heat; after about 15 minutes turn off one of the middle burners and turn the remaining burners down to medium. For charcoal grills, bank one chimney starter-full of lit and ashed-over charcoal briquettes to one side of the grill. Set up a drip pan on the other side to avoid flare-ups. (Be sure to consult the grill manufacturer's guide for best results.)
Lightly sprinkle the chicken all over with salt and pepper, and place its cavity over the cola can so that the chicken is sitting upright. Carefully put the chicken on the can on the indirect side of the grill, with the back facing the hotter side. Cover and cook until an instant-read thermometer inserted in the breast (avoiding the bone) reads 140 degrees F, 35 to 40 minutes. Baste the chicken all over with the glaze and cover for 5 minutes. Turn the chicken so that the breast is facing the hotter side. Baste a second time, and cover again until the thermometer reads at least 165 degrees F in the thickest part of the thigh, about 25 minutes more. Carefully remove from the grill and let rest 20 minutes before carving. Drizzle with some of the glaze and sprinkle with the sesame seeds. Serve the pickled peppers on the side.
All grills are different and you know yours best. Cooking times may vary, so adjust as needed.
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