Step 1

In a large bowl, combine the shredded Brussels sprouts, apples, shallots, olive oil, vegetable oil, apple cider vinegar, honey, salt, and pepper; toss well. Cover with plastic wrap and refrigerate for at least 30 minutes and up to 4 hours to allow the sprouts to soften and the flavors to marry.

Step 2

When ready to serve, toss the walnuts and Parmigiano-Reggiano with the salad. Taste and adjust seasoning if necessary, then serve. Leftovers keep well in the refrigerator for 1 or 2 days but keep in mind that the flavors will dull the longer the salad sits; add a little more cider vinegar and vegetable oil to perk it up.

Step 3

Note: Toasting the walnuts isn't essential, but it brings out their flavor and makes them crunchier. To toast them, preheat the oven to 350°F and set an oven rack in the middle position. Arrange the walnuts in a single layer on a baking sheet. Bake, checking frequently, until lightly toasted and fragrant, 6 to 10 minutes. Transfer immediately to a plate and let cool.