Bring 4 quarts of water to a boil in a large pot and add 1 tablespoon salt. Fill a large bowl with 2 quarts water, 2 cups ice, and 1 teaspoon salt.

Place the sugar snaps in the boiling water and cook until just tender, about 2 minutes. Remove the sugar snap peas from the boiling water and submerge it in the ice bath; leave it there until it is complexly chilled, about 2 minutes. Remove the sugar snaps from the ice bath and dry completely with a clean kitchen towel.

Place the sugar snaps in a clean bowl. Add the olive oil, three-quarters of the Pecorino Romano, the mint, and the basil. Mix to coat the sugar snaps evenly and season with salt and pepper.