In a small bowl, stir together the Buffalo sauce, brown sugar, chopped onion, and minced garlic.
Place the chicken wings in the basin of a 6-quart or larger slow cooker.
Pour the sauce mixture over the wings and stir well to ensure all the wings are evenly coated.
Cover and cook on LOW for 3 to 4 hours or on HIGH for 2 to 3 hours, until the chicken is tender and cooked through.
Set an oven rack about 6 inches from the broiler element and preheat the broiler to HIGH. Line a large baking sheet with foil.
Using tongs, carefully transfer the cooked wings from the slow cooker to the prepared baking sheet, arranging them in a single layer. Discard the remaining liquid in the slow cooker.
Brush the tops of the wings with additional Buffalo sauce.
Broil for 3 to 5 minutes, watching closely, until the sauce is bubbly and the skin begins to crisp and brown in spots. Flip the wings, brush with more sauce, and broil for another 3 to 5 minutes.
Confirm the wings have reached a safe internal temperature of 165°F (74°C) with a meat thermometer. Serve immediately with celery, carrots, and dressing.