Preheat oven to 450 degrees. Arrange drumsticks in a 9-by-13-inch baking dish. Rub all over with chili powder; season generously with salt and pepper. Bake until tender, 35 to 40 minutes.
Meanwhile, make barbecue sauce: In a small saucepan, whisk together chili sauce, vinegar, sugar, espresso powder (if using), and mustard. Bring to a boil; reduce to medium-low, and simmer 2 minutes. Divide sauce between two bowls: one for brushing and basting, and one for dipping.
Remove drumsticks from oven, and brush with sauce. Return to oven; bake, turning and basting halfway through, until well browned, 8 to 10 minutes. Brush drumsticks with any leftover pan juices. Serve with reserved dipping sauce.
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