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.