Whisk all vinaigrette together (in order) in a bowl. Set aside.
In a large nonstick saute pan over med-high heat, a enough olive oil to coat the pan. Add brussels sprouts and saute until the sprouts start to wilt a little and start to caramelize. Take off heat and toss in the vinaigrette and walnuts.
Transfer into a salad bowl and let slighty cool. Top with shaved parmesan and serve warm or room temp.