Cook sugar snaps in an 8-quart pot of boiling salted water for 2 minutes, then transfer 1 cup sugar snaps to a colander and rinse under cold water to stop cooking.
Transfer cooled sugar snaps to a cutting board.
Cook sugar snaps remaining in pot until tender, about 2 1/2 minutes more, then transfer with a slotted spoon to a bowl.
Measure out and save 1 cup cooking water, reserving remaining water in pot.
Return cooking water in pot to a boil and cook pasta until al dente, then drain in colander.
While pasta is cooking, cut 1 cup sugar snaps (on cutting board) crosswise into 1/2 inch pieces.
Puree half of sugar snaps from bowl, half of garlic paste, 2 tablespoons oil, 1/4 cup cheese, and 1/4 cup saved cooking water in a blender, then force puree with a rubber spatula through a medium-mesh sieve into a large bowl.
Puree another batch in same mannner, forcing through sieve into bowl, and add cut sugar snaps.
Toss hot pasta with sugar snap sauce and, if necessary, enough of remaining 1/2 cup saved cooking water to thin sauce to desired consistency, then season pasta with salt and pepper.