Place first 8 ingredients in a blender; process until smooth. Set aside.
Leave root and 1 inch of stem on beets; scrub with a brush. Place beets and sweet potato in a shallow baking dish. Bake at 400° for 45 minutes. Add mushrooms, and bake an additional 15 minutes. Cool to touch. Peel beets and potato; cut into 1/2-inch cubes. Thinly slice mushrooms.
Combine beets, potato, mushrooms, greens, and chicken. Drizzle dressing over salad; toss well. Serve immediately.