Cook beets in medium pot of boiling salted water until tender, about 15 minutes. Drain; cool. Peel beets and cut in half.
Melt 2 tablespoons butter in heavy large nonstick skillet over high heat. Add shallots and mushrooms; sauté 3 minutes. Add sugar snap peas; sauté 2 minutes. Mix in mustard seeds, thyme and sugar. Add 2 tablespoons butter, fennel and peas. Stir-fry 4 minutes. Add green onions, parsley, vinegar and beets; toss to blend. Season with salt and pepper.