Using both pistachio oil and roasted pistachios creates a rich, nutty flavor that pairs well with roasted beets. Serve leftover yogurt with crispy chicken thighs or roasted sweet potatoes.
Preheat oven to 375°F. Toss together beets, thyme, 1 tablespoon oil, and 3/4 teaspoon salt in a 13- x 9-inch roasting pan. Spread beets evenly in pan; pour 1/2 cup water into pan. Cover pan tightly with aluminum foil. Bake in preheated oven until beets are tender, about 1 hour. (Larger beets will take longer to cook.) Remove from oven. Reduce oven temperature to 350°F.
Remove beets from pan; let stand until cool enough to handle, about 10 minutes. Using a paper towel, rub each beet to remove skin; discard skins. Cut beets in half lengthwise. (Cut any larger beets into quarters.) While beets are still warm, transfer to a large bowl. Add orange juice, vinegar, remaining 1 tablespoon oil, and remaining 2 teaspoons salt; toss to coat. Let stand 15 minutes. Toss beet mixture; taste and adjust seasonings, if needed. Set aside until ready to serve.
Spread pistachios evenly on a small rimmed baking sheet. Bake at 350°F until lightly toasted, about 5 minutes. Transfer to a high-powered blender or food processor; pulse until finely chopped, about 10 times. Add warm water; process until mixture is the consistency of chunky peanut butter, about 1 minute and 30 seconds. With processor running, gradually drizzle in pistachio oil until mixture is mostly smooth and spreadable, about 1 minute. Transfer mixture to a bowl; fold in yogurt and salt.
Dollop 3 tablespoons yogurt mixture on each of 8 plates; top each with 1/2 cup beets. Garnish with chopped pistachios.