Toast the walnuts in a dry skillet over medium heat, stirring, until fragrant and a shade darker, 3–5 minutes; cool, chop, and transfer to a large salad bowl.
Place the radicchio in very cold water for 10 minutes to crisp and temper bitterness, then drain well and spin or pat completely dry; add to the bowl.
Whisk the lemon juice, Dijon mustard, honey, kosher salt, and black pepper in a measuring cup. Slowly drizzle in the extra-virgin olive oil while whisking until emulsified, then stir in the shallot.
Core and thinly slice the apples and add them to the bowl.
Shave the Parmigiano Reggiano and add it to the bowl.
Pour most of the dressing over the salad and toss until the leaves are glossy and evenly coated, adding more dressing as needed. Serve immediately while crisp.