Place garlic and salt in a food processor; pulse until finely minced, then process to a smooth paste, scraping down the sides as needed.

With the motor running, slowly stream in 3 cups oil, alternating with 1/2 cup lemon juice, a few tablespoons at a time, until the mixture is thick and fluffy. If the sauce breaks, add 1 tablespoon ice water and blend until smooth before continuing.

Spoon the sauce into a jar, swirl the top, and serve or refrigerate. Let it chill for at least an hour for a thicker texture.