Make the vinaigrette: Bring apple cider to a boil in a small saucepan over medium-high. Boil, stirring occasionally, until reduced to ¼ cup, 10 to 12 minutes. Pour into a small heatproof bowl; set aside, and let cool slightly, about 10 minutes.
Whisk vinegar, shallot, mustard, salt, and pepper into cider until well combined. Gradually whisk in oil, whisking constantly, until emulsified, about 2 minutes. Set aside.
Make the salad : Toss together radicchio, arugula, brussels sprouts, and salt in a large bowl until combined. Add walnuts, half of the cheese (about ½ cup), and ½ cup reserved vinaigrette; toss to coat.
Arrange prosciutto and apple on top of salad, and sprinkle with remaining half of shaved cheese (about ½ cup). Garnish with additional walnuts, and serve with remaining vinaigrette (about ¼ cup) on the side. Serve immediately.
