Preheat the oven to 350 degrees F. Add the apricots to a small saucepan and cover with water. Bring to a rapid simmer, then turn off the heat and let the apricots slightly soften, 3 to 5 minutes; strain and set aside.
Spread the pistachios on a rimmed baking sheet and toast until lightly browned and fragrant, 7 or 8 minutes; set aside.
Whisk the oil and vinegar in a medium bowl.
Shred the carrots on the large holes of a box grater. You can control the size of the shreds by changing the angle of the carrot; a more severe angle will create longer slices, whereas holding the carrot perpendicular to the holes will yield shorter, choppier pieces. Add the carrots and softened apricots to the dressing, season with salt and pepper and toss thoroughly to combine. Refrigerate the salad, covered, until the vinaigrette has been absorbed, up to 2 hours.
Chop the pistachios and add to the salad just before serving.
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