Bring a small pan of water to a boil; add garlic. Remove with a slotted spoon after 1 minute. Rinse under cold water; set aside. Add basil to boiling water; immediately remove and rinse under cold water. Reserve 1 tablespoon of cooking liquid. Transfer garlic and basil to a food processor, and add lemon juice, olive oil, reserved water, and the remaining 1/4 teaspoon salt. Process until smooth.