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Bring a large pot of salted water to a boil over high heat. Add the asparagus and cook until tender but not mushy, 3 to 4 minutes. Using a slotted spoon, transfer the asparagus to a strainer to drain. Separate 1/2 cup of the asparagus tips from the strainer and reserve. In a food processor, add the remaining asparagus and the garlic and pulse until finely chopped. Add the basil, pine nuts, cheese, lemon juice, and 5 tablespoons of the olive oil and pulse 12 times more. Season to taste with salt and pepper and transfer the pesto to a large bowl.
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Meanwhile, return the water to a boil and cook the pasta until al dente as directed on the package. Drain, reserving 1/4 cup of the pasta cooking water.
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To the bowl with the pesto, add the hot pasta and reserved pasta cooking water and asparagus tips and toss until the pasta is well coated with the pesto. Garnish with basil, pine nuts, and cheese, drizzle with the remaining 1 tablespoon olive oil, and serve.