Combine dressing ingredients in a mason jar or small container with a lid and shake well. Refrigerate while you prepare the salad.
Bring a large pot of water to a boil. Fill a large bowl about halfway with water and then fill with ice.
Place broccoli and cauliflower into the pot of vigorously boiling water and allow to boil for 2 minutes. Quickly drain and transfer to the bowl of ice water. Once vegetables are completely chilled, transfer to a colander to drain well. Pour off some of the water from bowl and replace with fresh ice. Repeat process with carrots, cooking about 3 minutes.
Combine all the vegetables in a large mixing bowl and toss with the dressing. Cover the bowl tightly with plastic wrap and refrigerate for at least two hours or overnight. When ready to serve, stir in Parmesan cheese.
Prep time does not include marinating time. Allow for at least two hours to marinate.