Remove bottoms and outer leaves from Brussels sprouts. Thinly slice using a knife, or cut using a food processor fitted with a thin slicing disk.
Melt butter in a large skillet over medium-high heat. Sauté onion 3 minutes or until it softens and starts to brown. Add sliced Brussels sprouts and walnuts. Continue cooking about 5 minutes or until sprouts wilt and brown.
Stir in cheese, salt and pepper and vinegar before serving.