In a 12-inch skillet over medium, toast the walnuts, stirring often, until fragrant and lightly browned, about 3 minutes. Transfer to a cutting board; reserve the skillet. Let the nuts cool slightly, then finely chop; set aside.

In a large saucepan, bring 2 quarts water to a boil. Add the pasta and 1 tablespoon salt, then cook, stirring occasionally, until al dente. Reserve ½ cup of the pasta cooking water, then drain.

In the same skillet over medium-high, combine the oil and garlic; cook, occasionally turning the cloves, until golden brown, 2 to 3 minutes.

Remove and discard the garlic, then add the radicchio, ½ teaspoon salt and ¼ teaspoon pepper; stir to coat.

Distribute the radicchio in an even layer and cook, stirring only once or twice, until well browned and tender-crisp, 2 to 4 minutes.

Add the pasta to the skillet and stir. Add half of the Parmesan, half of the walnuts and ¼ cup of the reserved pasta water.

Reduce to medium and cook, stirring constantly and vigorously, until the pasta is lightly sauced, 1 to 3 minutes; add additional pasta water as needed if the mixture looks dry.

Off heat, stir in the remaining Parmesan, half of the parsley and the lemon juice.

Taste and season with salt and pepper. Serve drizzled with additional oil and sprinkled with the remaining walnuts, the remaining parsley, generous grindings of black pepper and additional Parmesan.