Heat oven to 350° F. Spread the nuts on a rimmed baking sheet and toast in oven, tossing occasionally, until fragrant, 10 to 12 minutes. Rub the warm nuts between 2 clean dish towels to remove the skins (discard the skins).
Whisk together the oil, vinegar, mustard, and ½ teaspoon each salt and pepper. Add the radicchio, romaine, prosciutto, scallions, apple, and nuts and toss to combine. Serve immediately.