5 large cloves garlic, coarsely chopped (about 1/4 cup)
1 tsp. dried thyme
1/3 cup extra-virgin olive oil; more as needed
12 oz. cavatelli or similar pasta
1 lb. asparagus, cut into 1-1/2-inch pieces
1 large onion, finely chopped (about 1-1/2 cups)
1 lb. extra-large (26 to 30 per lb.) or large (31 to 35 per lb.) shrimp, shells removed
2 medium lemons, 1 zested and juiced, 1 cut into wedges for serving
Freshly ground black pepper
1/2 cup dry white wine or vodka
1 cup finely grated Parmigiano-Reggiano or Grana Padano; more for serving
1 cup chopped fresh flat-leaf parsley
1. Put the garlic and thyme in a small saucepan and add enough oil just to cover, about 1/3 cup. Warm the pan over very low heat so that the temperature of the oil is just below a simmer; you should see small bubbles rising from the oil at a slow, gentle rate. (If the oil begins to simmer, lower the heat. If you can’t reduce the heat any further, remove the pan from the heat to allow the oil to cool slightly, and then return it to the burner.) Continue to poach the garlic until it becomes slightly translucent (there should be no browning), about 20 minutes. Remove the pan from the heat and set aside.
2. Meanwhile, bring a large pot of well-salted water to a boil. Cook the pasta according to package directions until al dente, adding the asparagus in the last 2 minutes of cooking. Reserve 1 cup of the cooking water, and then drain the pasta and asparagus.
3. Heat 3 Tbs. of the garlic oil in a large skillet over medium-high heat. Add the onion and cook, stirring frequently, until translucent, about 2 minutes. Add the shrimp, lemon zest, 1/2 tsp. salt, and a few grinds of pepper, and cook, stirring frequently, until the shrimp begin to turn pink, about 11/2 minutes. Add the wine and cook, stirring occasionally, until the shrimp are pink and curled and most of the liquid has evaporated, about 3 minutes.
4. Add the pasta, asparagus, cheese, poached garlic, and garlic oil to the pan, toss to combine, and heat through, about 2 minutes. Remove from the heat, add the lemon juice and parsley, and season to taste with salt and pepper. If necessary, add some of the reserved cooking water to loosen the sauce. Divide evenly among six shallow bowls. Serve with lemon wedges, passing additional grated cheese.