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Preheat the oven to 400°F. Line a rimmed baking sheet with foil. Toss the asparagus on the prepared baking sheet with 2 tablespoons of the oil, then arrange close together in one layer. Sprinkle with salt and pepper. Roast until tender, about 12 minutes, shaking the pan a few times.
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Meanwhile, in a medium nonstick skillet over medium-high heat, warm 1 tablespoon of the oil. Add the prosciutto in one layer and cook until almost crisp, about 1 minute total, flipping halfway through. Drain on paper towels and break into bite-size pieces.
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Arrange the arugula on a serving platter or on 4 individual plates.
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In a small bowl, whisk the mustard, vinegar, lemon juice, 1 teaspoon salt, and ¼ teaspoon pepper. While whisking, slowly add the remaining 6 tablespoons oil until the vinaigrette is emulsified.
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Arrange the asparagus on the arugula and drizzle with enough dressing to moisten. Add a few egg quarters, some crispy prosciutto pieces, and Parmesan shavings and serve.