How to Make It
Bring apple cider vinegar and 2 tablespoons water to a boil in a small saucepan over high. Add raisins; boil 1 minute. Remove from heat; cool to room temperature. Drain.
Cook pasta according to package directions; drain and place in a large bowl.
Cut cabbage in half lengthwise (reserve 1 half for another use). Cut remaining half into 2 equal wedges. Cut whole head radicchio in half.
Heat a large cast-iron skillet over high. Add 1 tablespoon oil; swirl to coat. Add cabbage and radicchio, cut sides down, and cook until well charred, about 5 minutes. Remove radicchio from skillet. Turn cabbage wedges; cook until well charred, about 5 minutes. Remove from skillet.
Add 1 tablespoon oil to skillet; swirl to coat. Add onion wedges, cut sides down, and cook until well charred, about 4 minutes. Carefully turn wedges; cook until well charred, about 4 minutes. Remove from skillet.
Cut cabbage and radicchio crosswise into slices; separate onion wedges into “petals.” Add cabbage, radicchio, onion, balsamic vinegar, salt, pepper, and remaining 2 tablespoons oil to pasta; toss well to combine. Top each serving evenly with nuts, cheese, and pickled raisins.