Preheat your oven to 450°F (230°C), and line a baking dish with parchment paper or lightly grease it.
Place the peach halves, cut side up, in the prepared baking dish. Drizzle 1 tablespoon of avocado oil over the peaches and sprinkle with ¼ teaspoon of salt.
Bake the peaches in the preheated oven for 20-25 minutes, or until they are tender and slightly caramelized.
While the peaches are baking, tear the Burrata cheese into bite-sized pieces and set aside.
Once the peaches are done, remove them from the oven and let them cool slightly, then arrange the baked peach halves on a serving platter or individual plates. Top each peach half with pieces of torn Burrata cheese.
Drizzle the ½ tablespoon of olive oil over the peaches and Burrata, and then sprinkle them with the torn basil leaves and the remaining ¼ teaspoon of salt. Serve immediately, garnished with extra basil; add toasted or candied pistachios atop before serving, to add a bit of crunch and balanced sweetness.