Preheat oven to 425 degrees. Line a baking pan with foil or parchment paper. Set aside. Spread shredded Brussels sprouts out on the pan. Drizzle with olive oil and toss to coat. Sprinkle with salt.
Roast Brussels sprouts for 15-20 minutes depending on preference and thickness of your Brussels sprouts. When they're cooked, remove from oven and allow to cool slightly.
In a small bowl or a jar with a leak-proof lid, combine lemon juice, olive oil, grated garlic and salt. Shake vigorously until the mixture is emulsified -- it'll look a lighter, opaque yellow.
Toss together the roasted Brussels sprouts and arugula. Drizzle the mixture with vinaigrette and toss to coat. Sprinkle with Parmesan cheese before serving.