Position oven rack in the upper third of the oven; preheat broiler to high.
To prepare dipping sauce, combine orange zest, orange juice, water, honey, 1 tablespoon vinegar, and coriander in a small saucepan; bring to a boil. Whisk cornstarch and the remaining 1 teaspoon vinegar in a small bowl until smooth. Add to the saucepan and return to a boil, whisking until thickened, 30 seconds to 1 minute. Remove from the heat. Stir in mint.
Sprinkle drumsticks with salt and pepper
Coat a broiler pan with cooking spray. Sprinkle drumsticks with salt and pepper (Step 3), then broil, turning once, until an instant-read thermometer inserted into the thickest part registers 165°F, about 15 minutes total.
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