Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1/2 cup cooking water, then drain.
Meanwhile, whisk 3 tablespoons olive oil, the vinegar and beet juice in a large bowl; season with 1/2 teaspoon salt and a few grinds of pepper. Add the beets, radishes and radicchio and toss well to coat. Set aside.
Heat the remaining 5 tablespoons olive oil in a large skillet over medium-high heat. Add the red onion, garlic, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the onion is softened, about 4 minutes; add to the beet mixture and toss to coat.
Add the pasta, basil and 1/4 cup each ricotta salata and reserved cooking water to the beet mixture; season with salt and pepper. Toss well to coat, adding enough of the remaining cooking water to loosen. Top with the remaining ricotta salata.
Photograph by David Malosh
Courtesy of Food Network Magazine