Heat the oil in a large heavy-bottomed pot or Dutch oven over medium heat. Add the onion and cook, stirring occasionally, until softened and golden, 5 to 7 minutes. Add the garlic and chili powder and cook until the garlic is softened, about 1 minute. Stir in the ketchup, vinegar, mustard, brown sugar, 1 teaspoon salt and several grinds of pepper. Stir to combine.
Add the chicken to the sauce, turning to coat. Lower the heat to medium-low and keep at a steady simmer, adjusting the heat if necessary. Cover and cook, flipping the drumsticks halfway through, until the drumsticks are tender when pierced with a knife and the internal temperature registers 165 degrees F on an instant-read thermometer (avoid touching bone), 35 to 40 minutes. Remove the lid and continue to simmer until the sauce is as thick as molasses, turning the drumsticks several times to coat well and to keep from sticking, about 5 minutes. Remove to a plate and serve immediately.
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