Add all dressing ingredients to a container with a tightly fitting lid and shake to mix well. Let sit while cutting vegetables.
Cut all vegetables into 1 inch pieces. Shake dressing and pour half of it over the carrot and potato. Toss to coat. Spread over a baking sheet and put in oven. Set timer for 15 minutes.
While carrots and potatoes are baking, toss the more fragile vegetables in remaining dressing. Spread out in a separate baking pan.
After carrots and potatoes have cooked for 15 minutes, remove from oven and stir. Put both pans back in oven and cook for an additional 20 - 25 minutes, removing the pans half way through to stir again. If mushrooms are releasing liquid, drain it from the pan before putting back into the oven.
Once veggies are all cooked through, remove from oven, toss together, and serve.