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To make the vinaigrette: In a small saucepan, bring the apple cider to a boil and cook until reduced to 1/4 cup.
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Let cool completely.
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In a medium bowl, combine the vinegar, shallots, mustard, and reduced apple cider.
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Whisk well and continue whisking while adding the oil in a slow, steady steam stream until emulsified.
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Season to taste with salt and pepper.
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The vinaigrette can be transferred to a tightly closed jar and refrigerated for up to 1 week.
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(Shake well to reemulsify before using.
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).
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To make the salad: in a large bowl, toss the squash with just enough oil to lightly coat and season with salt and pepper.
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Heat a large nonstick skillet over medium heat.
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Working in batches, add the squash in a single layer, cover and cook, stirring occasionally, until tender, about 5 minutes.
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Uncover and cook, tossing occasionally, until browned, about 2 minutes longer.
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Transfer to a plate.
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Repeat with the remaining squash.
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Let cool.
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In a large bowl, combine the greens, apples, and squash.
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Gently toss with just enough vinaigrette to lightly coat.
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Any leftover vinaigrette can be refrigerated for up to 1 week.
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Divide among six serving plates and top with the cheese and walnuts.
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For the walnuts:.
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Preheat the oven to 300F Line a half sheet pan with a Silpat or other nonstick silicone baking mat, or parchment paper.
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In a large bowl, beat the egg white until soft peaks form.
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Add the walnuts and toss gently until well coated.
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In a medium bowl, combine the sugars, ginger, salt, and pepper.
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Sprinkle over the walnuts and gently fold until the nuts are evenly coated.
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Spread in a single layer on the pan.
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Bake, stirring every 10 minutes, until golden brown, about 45 minutes.
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Transfer the pan to a wire rack.
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Separate the nut with a fork and let cool completely in the pan.
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The cooled nuts will keep in an airtight container for up to 1 week, but are best when fresh.
