Bring the beets to a boil in a saucepan with salted water. Simmer and cook until fork tender - about 20 - 30 minutes. Allow to cool, peel and cut into bite-sized pieces.
In a large bowl, whisk vinegar, salt and pepper, to taste. Whisk in the olive oil in a slowly to make a dressing. Toss the cut beets in the dressing and set aside to marinate for at least 30 minutes or up to 2 hours.
In a dry skillet, toast the nuts. Stir to avoid over-browning, and remove from heat when they are golden brown. Cool.
Toss the arugula and spinach or spring mix with the beets and divide among 4 plates. Sprinkle the walnuts and goat cheese on top.