Wash brussels sprouts; remove discolored leaves. Cut off stem ends and thinly slice brussels sprouts. (They should look shredded.)
Heat butter and oil in a large deep skillet or Dutch oven over medium-high heat until hot. Add shredded brussels sprouts, garlic, and onion slivers. Saute 8 to 10 minutes or until brussels sprouts are tender and onion is lightly caramelized. Season with salt and pepper; transfer to a serving bowl. Add vinegar and brown sugar to skillet. Simmer over medium heat 30 seconds; pour over brussels sprouts, and toss gently. Serve hot.