Preheat oven to 425°F. Line a large sheet pan with parchment paper or foil for easy cleanup.
In a large bowl, whisk together BBQ sauce, oil, garlic, soy sauce, apple cider vinegar, brown sugar, smoked paprika, onion powder, chili powder (if using), salt, and black pepper until smooth.
Add chicken drumsticks and coat well. Cover and marinate for at least 1 hour or overnight in the fridge for best flavor.
Arrange marinated drumsticks on the sheet pan in a single layer.
Bake chicken for 20 minutes to start rendering fat and building flavor.
While chicken bakes, lightly toss corn on the cob and green beans with a small drizzle of oil, salt, and pepper.
After 20 minutes, carefully remove the pan from the oven.
Add corn and green beans around the chicken in a single layer.
Return to the oven and bake for another 20–25 minutes, until chicken is fully cooked (internal temp 165°F / 74°C), corn is tender, and green beans are roasted and slightly crisp.
Brush chicken with extra BBQ sauce during the last 10 minutes of cooking for a sticky glaze.
Optional: Broil for 3–5 minutes at the end for charred edges and extra caramelization.
Let rest for 5 minutes before serving.