Step 1

Preheat oven to 400 degrees F. Place a large piece of foil on a rimmed baking sheet; place beets in the center of the foil. Gather the sides of the foil up to make a packet; pour in 1/2 cup vinegar and seal the packet. Roast until the beets are very tender when pierced with a knife, about 45 minutes. Open the foil packet; let cool for 30 minutes.

Step 2

Meanwhile, bring the remaining 1/2 cup vinegar to a boil in a small saucepan over medium-high heat. Cook, without stirring, until reduced to about 3 tablespoons, about 10 minutes. Remove from heat (the mixture will thicken as it cools).

Step 3

When the beets are cool enough to handle, peel them, then slice very thinly with a knife or a mandoline. Arrange the sliced beets, overlapping slightly, on a platter.

Step 4

Toss arugula with oil, salt and pepper in a medium bowl; arrange on top of the beets. Sprinkle with goat cheese and walnuts; drizzle with the balsamic reduction.