Preheat the oven to 425 degrees F. Place the beets on a sheet of foil. Drizzle with olive oil and season with salt. Wrap the foil around the beets and place on a baking sheet. Roast until tender when pierced with a knife, 50 to 60 minutes. Let cool slightly, then unwrap and rub off the skins using a paper towel.
About 30 minutes before serving, remove the burrata from the refrigerator and let stand at room temperature. Whisk the orange juice, vinegar and shallot in a small bowl. Slowly whisk in the 2 tablespoons olive oil until emulsified.
Thinly slice the beets and arrange in a single layer on a serving plate, overlapping slightly. Spoon the vinaigrette over the beets and season with 1/2 teaspoon salt and a few grinds of pepper; set aside.
Preheat the broiler. Arrange the bread on a baking sheet and broil until toasted. Sprinkle the beets with the chives and pistachios. Place the burrata on the beets. Drizzle with olive oil and season with salt and pepper. Serve with the toasted bread.