Bring 3 quarts of water to a boil and add 1 tablespoon salt. Set up an ice bath nearby. Blanch the asparagus spears in the boiling water for 1 minute and 30 seconds, then remove and immediately refresh in the ice bath. Once cooled, remove the asparagus from the ice bath with tongs and drain slightly. Set aside.
In a 12 to 14-inch saute pan, heat 3 tablespoons of olive oil over high heat. Add the asparagus and toss in the oil over high heat for 1 minute. Season with salt and pepper, then divide evenly among 4 warmed dinner platters.
In a separate pan, add the remaining 3 tablespoons olive oil, and heat over medium-high heat. Crack the eggs into the pan, taking care to keep the yolks intact. Cook the eggs sunny-side up until the whites are firm but the yolk is still runny. Season each egg with salt and pepper, the carefully slide 1 egg onto each serving. Grate the cheese over each plate, season with a squeeze of lemon juice and salt and pepper. Serve immediately.
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