Preheat your oven to 400 degrees F. Line a large rimmed baking sheet with foil.
Spray the drumsticks all over with oil. Place them on the prepared baking sheet and bake them until cooked through, 30-45 minutes, depending on their size and on how hot your oven runs. Aim for an internal temperature of 165F when not touching the bone.
Make the sauce:
10 minutes before the drumsticks should be done, prepare the teriyaki glaze. In a small saucepan, whisk together the soy sauce and cornstarch.
Add the white wine, honey, minced garlic, and minced ginger, whisking to combine.
Heat the sauce over medium-low heat, whisking often, until it thickens into syrup. This should take 3-4 minutes. Remove from heat.
Glaze the chicken:
Remove the baked chicken pieces onto a serving platter. Brush them with the teriyaki glaze, sprinkle with sesame seeds, and serve.
Replace the drumsticks with chicken thighs. If you use boneless chicken thighs, you'll probably need to bake them for about 30 minutes. For large bone-in thighs, try 40-45 minutes.
I keep the skin on, but you can remove it if you wish. Both versions are very good. If you do remove the skin, be generous when coating the chicken pieces with oil to help prevent them from drying out.