Put the beets in a saucepan with water to cover and season generously with salt. Bring to a boil over high heat and cook until fork tender, about 20 minutes. When the beets are cool enough to handle, peel them--the skins should slide right off with a bit of pressure from your fingers. If they don't, use a paring knife to scrape off any bits that stick. Cut each beet into bite-sized wedges.
Whisk the vinegar with salt and pepper, to taste, in a large bowl. Whisk in the olive oil in a slow steady stream to make a dressing. Toss the cut beets in the dressing; set aside to marinate for at least 15 minutes or up to 2 hours.
Preheat the oven to 350 degrees F. Spread the nuts on a baking sheet and oven-toast, stirring once, until golden brown, about 8 minutes. Cool.
Toss the arugula and escarole with the beets and divide among 4 plates. Scatter the walnuts and goat cheese on top. Serve.
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