Toast the walnuts first. Spread them on a dry baking sheet and bake at 350 degrees for 6 to 8 minutes, stirring once halfway through, until they smell nutty and look a shade darker. Watch them closely because they go from toasted to burnt fast. Pour onto a plate and let them cool completely.

Make the vinaigrette. In a small jar or bowl, combine the cider vinegar, Dijon mustard, honey, salt, and pepper. Whisk or shake to blend. Add the olive oil and whisk until the dressing looks smooth and slightly thick. Taste it on the tip of your finger. It should be tart, a little sweet, and bright. Adjust salt if needed.

Slice the fennel. Cut each bulb in half through the root, then slice each half as thin as you can on a mandoline set to 1/8 inch, or use a very sharp knife and take your time. You want the slices thin enough that they almost curl. Drop them into a large mixing bowl with the cold water and a squeeze of lemon if you are working ahead, or go straight to assembling.

Slice the apples. Core them and cut into thin half-moons, about 1/8 inch thick. If you are assembling right now, add them directly to the bowl. If you are waiting even 10 minutes, toss the apple slices with a little lemon juice to keep them from browning.

Add the arugula to the bowl with the fennel and apples. Drizzle about two-thirds of the vinaigrette over everything and toss gently with your hands or two spoons so every slice gets coated. Taste a piece of fennel. If it needs more dressing, add the rest.

Transfer to a serving platter or wide shallow bowl. Scatter the toasted walnuts over the top, then lay the Parmesan shavings across the salad. Finish with a few of the reserved fennel fronds for color and a pinch of flaky salt if you have it. Serve immediately.