Bring a large pot of salted water to a boil. Cook the pasta according to the package instructions. Reserve ½ cup water before draining. Transfer the pasta to a large bowl.
In a small pan, heat the olive oil over medium heat. Add the shallots and sauté for 4 minutes until soft.
Transfer the shallots to a blender. Add the beets, feta, parmesan, garlic, salt, and pepper. Blend until smooth, adding ¼ to ½ cup of pasta water to create smooth sauce.
Pour the sauce over the pasta and toss well.
Serve with a garnish of pistachios, fresh parsley, and parmesan.