Place brown sugar, soy sauce, sake or sherry, and garlic in a large resealable plastic bag and stir to combine.
Add chicken, seal (pressing out excess air), and turn the bag to coat the chicken evenly.
Place in the refrigerator and marinate, turning occasionally, for at least 2 hours or up to 12 hours.
Heat the broiler to high and arrange a rack in the middle of the oven. Line a baking sheet with aluminum foil and set a wire rack over the foil.
Remove the chicken from the marinade, letting any excess drip off, and arrange in a single layer on the rack. (Try not to let the wings touch.) Discard the marinade.
Broil the chicken until the meat starts to pull away from the bones and the skin is a deep golden brown, bubbling, and crisped, about 12 minutes. Remove from the oven and, using tongs, flip the wings over. Return to the oven and continue broiling until the skin is a deep golden brown, bubbling, and crisped on the second side, about 12 minutes more.
Meanwhile, prepare the sauce.
For the sauce:
Place all ingredients except cornstarch-water mixture in a small saucepan and stir to combine.
Bring to a boil over medium-high heat.
Reduce heat to low and simmer until sauce has reduced to about 1/2 cup, about 10 minutes.
Add cornstarch mixture, whisk to combine, and cook until sauce has thickened, about 1 minute.
Transfer to a large heatproof bowl.
Add cooked wings to the sauce and toss to combine.
Transfer to a serving dish and sprinkle with toasted sesame seeds. Serve immediately.