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Preheat the oven to 350°F.
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Toast the hazelnuts on a baking sheet until lightly golden brown, shaking the pan occasionally to jostle the nuts and ensure they cook evenly, about 10 minutes.
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Cool completely.
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Rub the hazelnuts between your palms to remove the dark skins from the nuts.
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Very coarsely chop the hazelnuts.
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Blend the shallot, mustard, garlic, and 2 tablespoons of the vinegar in a blender until very finely chopped.
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Add the remaining 2 tablespoons of vinegar and blend until smooth.
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With the blender on low, slowly drizzle in the oil until emulsified.
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Season the vinaigrette to taste with salt and pepper.
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Toss the arugula in a large bowl with enough vinaigrette to coat.
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Season to taste with salt and pepper.
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Mound the salad on 4 plates, dividing equally.
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Sprinkle with the hazelnuts and serve.