Preheat oven to 350 degrees. Spread nuts evenly on a rimmed baking sheet; toast in oven until golden and fragrant, tossing once, 8 to 10 minutes. Let cool completely.
Meanwhile, bring 4 cups salted water to a boil; add basil, and submerge with a spoon. Immediately drain in a colander. Rinse with cold water until cool, then pat basil completely dry in paper towels.
In a food processor, combine nuts, basil, and garlic; season generously with salt and pepper. Process until nuts are finely chopped. With machine running, pour oil in a steady stream through the feed tube; process until smooth. Use immediately, or freeze.