Preheat oven to 375 degrees F (190 degrees C). Wrap each beet in two layers of aluminum foil, and place onto a baking sheet. Bake in the preheated oven until the beets are tender, about 1 hour and 20 minutes. Allow the beets to cool slightly, then remove the skins. Let the beets cool to room temperature, or refrigerate until cold. Once cooled, thinly slice the beets.
Place the mache and arugula into a large mixing bowl. Add the sliced beets and peaches; sprinkle with the shallots, pistachios, and goat cheese. In a separate bowl, whisk together the walnut oil, balsamic vinegar, salt, and pepper until emulsified, and pour over the salad mixture. Toss well, and serve.