Heat the oven to 425ºF. Put the beets in a small baking dish and add water to come ½ inch up the sides. Cover tightly with foil and roast until a knife slides easily in and out of the beets, 50 to 60 minutes.
Spread the kale on two rimmed baking sheets and drizzle with 2 tablespoons of the oil. Using your hands, toss to coat the leaves with the oil, then sprinkle on ¼ teaspoon of the salt and ⅛ teaspoon of the pepper. In the last 10 minutes of the beets’ cooking time, place the kale in the oven and roast until the edges brown.
Remove the beets and kale from the oven and let them cool in the pans uncovered. When they are cool enough to handle, cut off the ends of the beets and slip off the skins. Cut the beets into ¼-inch-thick slices and set aside.
While the beets are cooling, squeeze 2 tablespoons of juice from the orange pieces into a medium bowl. Gradually whisk in 4 tablespoons of the oil until thickened. Whisk in ¼ teaspoon of the salt and ⅛ of the teaspoon pepper. Set the vinaigrette aside.
Season the chicken with the thyme, ½ teaspoon of the salt and ⅛ teaspoon of the pepper. Heat the remaining 1 tablespoon of oil in a large nonstick skillet over medium-high heat. Add the chicken and cook until browned on both sides and the internal temperature reaches 165ºF, 4 to 5 minutes per side; remove to a cutting board to cool for 5 minutes. While warm, slice the chicken on a diagonal and toss it in the vinaigrette to coat.
Arrange the kale and beets on four plates. Sprinkle the beets with the remaining ¼ teaspoon of salt and ⅛ teaspoon of pepper. Top with the chicken, remaining oranges, and pistachios. Drizzle on the remaining vinaigrette.
Tip: The beets can be roasted two days ahead of time and kept refrigerated. Bring them to room temperature before using. If your beets have the tops attached, remove them, chop the greens, and use them in place of some of the kale.