Bring 2 qt. water seasoned with 1 Tbs. of the salt to a rolling boil. Meanwhile, set up an ice bath by combining ice and water in a medium bowl.
Put the basil in the boiling water, pressing it gently under the water, and cook for just 2 or 3 seconds. Quickly remove the basil from the water (a Chinese strainer works great here) and plunge it immediately into the ice bath to stop the cooking process. Let cool completely for 2 minutes.
Remove the basil from the ice bath and squeeze it lightly with your hands or in a clean dishtowel to remove most of the excess water.
Chop the basil coarsely with a sharp knife and then put it in a blender. Add the 1/4 cup ice water, the garlic, the pine nuts, the cheese, the remaining 1 tsp. salt, and the pepper. Blend until the basil is coarsely puréed, scraping down the sides and adding more water to facilitate blending only if needed. Be patient; don’t add more water if it isn’t necessary. With the blender running, add the oil in a steady stream until the pesto looks creamy and emulsified. Cover and store in the refrigerator for a few days or in the freezer for up to a few months.
Chopping the basil speeds blending. Squeeze out the excess water first.