Cook beets in large saucepan of boiling water until tender, about 45 minutes. Drain. Cool. Peel beets. Cut beets into thin wedges. Place in medium bowl.
Whisk vinegar, shallots and fennel seeds in small bowl. Gradually whisk in olive oil. Season to taste with salt and pepper. Drizzle 1/4 cup vinaigrette over beets; toss to coat.
Cook quinoa in large pot of boiling salted water until tender, about 10 minutes. Drain well. Rinse with cold water and drain again. Place quinoa in large bowl. Add cheese, fennel and arugula and mix gently. Add remaining dressing and toss to coat. Transfer salad to large bowl. Arrange beets atop salad.
c.1996, M.S. Milliken & S. Feniger, all rights reserved