Step 1
Bring 4 quarts of water to a boil in a stockpot. Lightly salt and add the pasta, stir to separate and cook according to the package instructions, until al dente. Drain and return the pasta to the pot.
Step 2
While the pasta is cooking, in a 12-inch nonstick skillet over high heat, heat 2 tablespoons of oil until it just begins to smoke. Add the asparagus, pepper and the 1/2 teaspoon of salt. Toss to coat and then spread the asparagus in a single layer and cook without stirring, until it begins to brown on one side, about 1 minute.
Step 3
Add the walnuts and continue to cook, stirring frequently, until the asparagus is tender-crisp and the nuts are toasted, about 4 minutes. Add the arugula and toss until just starting to wilt, about 1 minute.
Step 4
Add the asparagus mixture, cheese, vinegar and the remaining 3 tablespoons of oil to the pasta and toss to combine.
Step 5
Divide the pasta among bowls, and use the coarse side of a box grater to grate a little apple over each bowl. Serve hot.
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