Cut off the asparagus tips (reserving them). Then, using a sharp vegetable peeler and holding each spear by its woody ends against a cutting board, peel the stalks into long, thin strips. Discard the woody ends.
Bring a large pot of water to a boil over medium-high heat. Add the pasta and cook according to the package directions, until al dente. Drain and return the pasta to the pot.
Meanwhile, add 2 tablespoons oil and the garlic to a large, deep skillet over medium heat; cook for about 5 minutes, stirring, until the garlic is lightly golden, being careful not to let it burn.
Add the wine (be careful, it will sputter at first) and the stock; bring to a boil over medium-high heat and cook for about 5 minutes, until the liquid has reduced by about half.
Add the asparagus tips and strips, the lemon zest, lemon juice, salt and pepper. Let the mixture start to bubble again, then cook for about 2 minutes, just until the asparagus is crisp-tender. Stir in the crabmeat and cook for about 1 minute, just long enough for it to warm through.