Preheat the oven to 400 degrees. Prepare a large sheet pan or two regular-sized sheet pans with aluminum foil spritzed with nonstick cooking spray or rubbed with olive oil.
In a bowl, whisk together the soy sauce, honey, chili garlic sauce, garlic powder, ginger, and cornstarch.
Place the chicken drumsticks in a gallon-size zip-top freezer bag. Pour all but ½ cup over the chicken. Seal the bag and allow it to rest on the counter while the vegetables are prepared.
Dump the bag of chicken on the prepared sheet pan.
Bake for 15 minutes. Carefully remove the baking sheet from the oven and add the vegetables. Arrange the vegetables around the chicken legs. Pour the remaining sauce over the vegetables and chicken.
Return the sheet pan to the oven and bake for an additional 30 minutes.
If desired, turn on the broiler and cook for an additional 2 to 5 minutes, watching closely.