For the dressing, whisk together the olive oil, cider vinegar, honey, kosher salt, and fresh ground black pepper.
If desired, toast the walnuts by placing them in a dry skillet over low heat for several minutes, stirring frequently, until slightly browned and fragrant. Immediately remove from the heat into a bowl.
Thinly slice the cabbage. Core 2 apples and chop them. (If not eating immediately, sprinkle the apples with a bit of lemon juice to prevent browning.) Using a vegetable peeler, shave the Parmesan or Pecorino Romano cheese.
To serve, place cabbage on a serving plate, then top with apples, walnuts, cheese, and dressing.
