Roast the Beets: Preheat the oven to 375°F (190°C) with a rack in the center position. Trim the beet greens (reserve and use for sauteed beet greens), leave roughly ½-inch stem visible. Do not remove or trim the root end. Scrub and clean the beets gently, without piercing or removing the skin.
Wrap each beet tightly in aluminum foil, and place them in a roasting pan and wrap tightly with foil. Bake for at least 45 minutes or up to 1¼ hours. Time will vary depending on the size of beets. Check the beets by unwrapping them, be very careful for steam, and piercing the center of the beet with a paring knife. If the knife meets no resistance, the beets are finished cooking. Set aside to cool completely.
Wait until the beets are cool enough to handle (they will be easier to peel when slightly warm). Peel gently with fingers, remove root and tail, and slice into half-moons or wedges.
Prepare the Candied Pistachios: Meanwhile, combine the pistachios in a small bowl with the granulated sugar, 1 teaspoon of water, and a pinch of salt. Place the pistachios on a half sheet pan, sprinkle with the turbinado sugar, and toast for 10 to 12 minutes in a 375°F (190°C) by themselves or with the beets as they are cooking, or until the pistachios are caramelized and golden. Set on a rack and allow the pistachios to cool completely.
Prepare the Peaches: Cut a very thin (just enough to pierce the skin) “X” on the bottom of each peach. Place the peaches in a big bowl and cover completely with hot boiling water. Allow the peaches to blanch in the hot water for at least 5 minutes, or until the peel is easily removable. Right before serving, slice the peaches in half, core, and cut them into wedges.
Assemble: Drizzle a large platter with basil oil. Distribute the peach and beet slices amongst the plates. Set the burrata on a paper-towel lined plate to absorb any excess liquid, cut in half, and set on the platter. Add more basil oil as desired. Drizzle the plate lightly with high-quality olive oil and sprinkle with flaky salt. Garnish with chopped candied pistachios and serve immediately.
Tips for Success:
Roasted and peeled beets can be prepared up to 2 days in advance, and kept in a container in the fridge. Feel free to halve the basil oil recipe; however it will last in the fridge for 3 to 4 weeks (and can be used for a variety of other dishes!). Candied pistachios can be prepared several days in advance.