Cut stems off beets and place whole on a baking sheet. Roast until fork tender, about 45 minutes. Allow to cool; peel and dice beets into large bite-sized pieces. Toss with 1 tablespoon olive oil and 1 tablespoon balsamic vinegar in a bowl; season with salt and black pepper.
Toss diced avocado with 2 tablespoons olive oil in a separate bowl; season with salt.
To serve, divide salad greens between serving plates. Top each serving with 1/4 of the beets and avocado.
Sprinkle each serving with shelled pistachios, dried cherries, and shredded Parmesan cheese.
Drizzle with remaining 1 tablespoon olive oil and 1 tablespoon balsamic vinegar. Season with salt and ground black pepper to taste.