Add the garlic cloves and salt to a food processor and pulse until the garlic is finely chopped and turning into a paste.
With the motor running, very slowly drizzle in the oil a drop or 2 at a time. Once 1/4 cup of the oil is added this way, you can drizzle it in a thin stream instead of drop-by-drop.
After 1/2 of the oil is added this way, drizzle in the lemon juice with the motor running. Once 1/2 of the lemon juice is added, alternate between drizzling in lemon juice and oil.
Continue drizzling in the remaining oil the same way until it’s all incorporated.
With the motor running, drizzle in the ice water. The sauce will thin out and get super creamy.
Serve, or store in an airtight container in the fridge for up to 1 month.