Scrub the beets very well. Fill a 3-quart saucepan halfway with water and bring to a boil. Cook the beets until tender when pierced with sharp knife, about 45 to 60 minutes. Drain well and let cool. Slip the skins off the beets. Dice the beets and place in a bowl.
To make the dressing: whisk together the lemon juice, garlic, salt, pepper, and oil. Pour a few tablespoons of the dressing on the beets and toss.
Just before serving, combine the greens, arugula, and scallion in a large bowl. Pour on most of the remaining dressing and toss. Add more if needed.
Place the greens on 4 salad plates. Spoon on a mound of beets. Sprinkle with goat cheese and walnuts. Serve immediately.