Preheat the oven to 400 degrees F.
Place the beets on a large piece of aluminum foil. Drizzle with 1 tablespoon of the olive oil and sprinkle generously with salt and pepper. Place the orange peels among the beets. Fold the aluminum foil up and around the beets to create a closed packet.
Place the packet on a baking sheet and roast until the beets are tender and the edges of the orange peels begin to caramelize, 45 to 50 minutes. Open the packet carefully and discard the orange peels. Allow the beets to cool slightly, about 10 minutes.
Meanwhile, prepare the components of the salad: Slice the peeled oranges into 1/4-inch-thick rounds. Use a horizontal peeler to make long, thin strips of the cucumber from end to end. Make the dressing by whisking the Greek yogurt with the mint and lemon zest and juice in a small bowl. Season the dressing with salt and pepper. Set aside.
Once the beets have cooled slightly, use the edge of a paring knife to peel off the skins. Slice the beets into 1/4-inch-thick rounds.
To assemble the salad, arranges the sliced beets in a shingled pattern on a serving platter. In between every 2 to 3 slices of beets, slip a round of orange into the mix. Using a small spoon, dollop the yogurt dressing onto the salad. Curl the cucumber ribbons over the surface of the salad. Drizzle the platter with the remaining 2 tablespoons olive oil. Sprinkle with the pistachios and pomegranate seeds. Serve chilled or at room temperature.