In a heavy Dutch oven or pot, mix the soy sauce, water, honey, brown sugar, ginger, and garlic together. Nestle the chicken thighs in the sauce, skin side up, submerging the meat as much as possible. Bring the mixture to a boil over medium-high heat. When the sauce comes to a boil, add the Maui or yellow onion wedges and green onion pieces and turn the heat to medium-low. Cover the pot with a lid and let simmer for 30 minutes. Using a pair of kitchen tongs, gently turn all the chicken pieces over. Cover the pot again and simmer for another 30 minutes.
Check the tenderness of the meat with a fork; if you can easily insert the fork into the meat, the chicken is done. Remove the cooked thighs from the sauce, reserving it, and place them on a rimmed baking sheet.
Preheat the broiler.
In a small bowl, whisk the cornstarch with 1/4 cup of the sauce until smooth, then add the cornstarch mixture back to the pot with the remaining sauce and cook over medium-low for 4 to 5 minutes, until the sauce has thickened. Broil the chicken thighs for a minute or two, watching carefully to make sure you do not burn the skin. The goal is to just quickly brown the skin.
Serve the chicken with the thickened sauce and rice.