Step 1

In a large saucepan of salted boiling water, cook the pasta until almost al dente, 5 to 7 minutes. Add the asparagus and cook until the pasta is al dente and the asparagus is crisp-tender, about 2 minutes longer. Reserve 1 cup of the cooking water, then drain well.

Step 2

Meanwhile, in a very large skillet, melt the butter with the basil sprigs over moderate heat. Add the pasta and asparagus, the 1/4 cup of cheese and 1/2 cup of the cooking water. Cook over moderate heat, tossing, until the pasta is coated in a light sauce, 3 to 5 minutes; add more of the cooking water if needed. Season the pasta generously with salt and pepper. Gently fold in the crab and cook until hot, 1 to 2 minutes. Discard the basil sprigs. Garnish with basil leaves and serve, passing more cheese at the table.