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1
Position a rack in the center of the oven and heat to 425F.
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2
Cut the potatoes lengthwise into 1/4-inch-thick slabs.
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3
Mound on a large rimmed baking sheet, drizzle with 1 tablespoon of the oil and 1/4 teaspoon salt, and toss to coat the potatoes evenly.
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4
Arrange in a snug single layer and roast until just tender and starting to brown, 20 to 25 minutes.
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5
While the potatoes are roasting, prepare the squash.
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6
Trim the squash, halve them lengthwise, and scoop out and discard the seeds and strings.
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7
Slice into thin half-moons, about 1/8 inch thick.
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8
Transfer to a large bowl, drizzle with 2 tablespoons of the olive oil, and sprinkle on 1/4 teaspoon salt.
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9
With your hands, gently toss to coat evenly.
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10
When the potatoes are out of the oven, line 2 large rimmed baking sheets with parchment or a nonstick liner and arrange the squash slices across them.
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11
Roast, rotating the pans after 10 minutes, until the slices are just tender and starting to brown, 20 to 25 minutes.
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12
Be sure not to overcook, or the squash will dry out.
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13
Note: Some slices will be darker than others even when cooked perfectly; this provides textural interest and depth of flavor.
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14
When done, set aside and let cool to room temperature.
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15
In a small bowl, combine the shallot, lemon juice, vinegar, mustard, honey, and 1/8 teaspoon salt.
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16
Whisk to blend, and slowly drizzle in the remaining 1/2 cup olive oil, continuing to whisk vigorously.
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17
Just before serving, put the potatoes and squash in a large bowl and drizzle with the vinaigrette.
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18
Toss well with your hands, taste, and adjust with more vinegar, lemon juice, or salt as needed.
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19
Add the arugula, half the pomegranate seeds, and half of the Parmigiano and gently mix in with your hands.
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20
To serve, sprinkle the salad with the remaining cheese and pomegranate seeds and top with a few grinds of black pepper.
