1.    Position a rack in the center of the oven and heat to 425°F. Cut the potatoes lengthwise into ¼-inch thick slabs. Mound on a large rimmed baking sheet, drizzle with 1 tablespoon of the oil and ¼ teaspoon salt, and toss to coat the potatoes evenly. Arrange in a snug single layer and roast until just tender and starting to brown, 20 to 25 minutes.

2.    While the potatoes are roasting, prepare the squash. Trim the squash, halve them lengthwise, and scoop out and discard the seeds and strings. Slice into thin half-moons, about ⅛-inch thick. Transfer to a large bowl, drizzle with 2 tablespoons of the olive oil, and sprinkle on ¼ teaspoon salt. With your hands, gently toss to coat evenly.

3.    When the potatoes are out of the oven, line two large rimmed baking sheets with parchment or a nonstick liner and arrange the squash slices across them. Roast, rotating the pans after 10 minutes, until the slices are just tender and starting to brown, 20 to 25 minutes. Be sure not to overcook, or the squash will dry out. Note: some slices will be darker than others even when cooked perfectly; this provides textural interest and depth of flavor. When done, set aside and let cool to room temperature.

4.    In a small bowl, combine the shallots, lemon juice, vinegar, mustard, honey, and ⅛ teaspoon salt. Whisk to blend, and slowly drizzle in the remaining ½ cup olive oil, continuing to whisk vigorously.

5.    Just before serving, put the potatoes and squash in a large bowl and drizzle with the vinaigrette. Toss well with your hands, taste, and adjust with more vinegar, lemon juice, or salt as needed.

6.    Add the arugula, half the pomegranate seeds, and half of the Parmigiano and gently mix in with your hands. To serve, sprinkle the salad with the remaining cheese and pomegranate seeds and top with a few grinds of black pepper.