Drizzle beets with oil and season with salt. Wrap tightly in foil and roast until tender, about 1 hour. Peel and roughly chop once cool.
Pulse beets, walnuts, and tomatoes in a food processor. Season with salt and a pinch of red pepper flakes. You are essentially making a beet/walnut pesto, so look for a pesto like consistency.
Cook pasta in salted water according to package directions. Drain, reserving 1 cup of pasta water, and return to pot.
Toss pasta with beet mixture, adding pasta water as necessary until creamy.
Serve with ricotta, a drizzle of oil, and red pepper flakes.