In a large pot, cook the pasta until al dente in salted water according to the instructions on the package.
While the pasta is cooking, in a food processor add the beetroot, feta, olive oil, lemon juice, garlic, chili flakes and season with salt and pepper. Blitz everything until smooth.
Drain the pasta and set aside about half a cup of the pasta water.
Toss the pasta with the beetroot sauce and add some of the reserved pasta water. Toss until the pasta is all covered with the sauce.
Plate the pasta, sprinkle some crumbled feta cheese and crushed pistachios on top.