Add the peeled garlic cloves and kosher salt to a food processor. Process until the garlic forms a smooth paste, stopping to scrape down the sides as needed.
With the processor running, begin to drizzle the vegetable oil in very slowly: start with just a couple drops at a time until about 1/4 cup is incorporated, then continue a very slow steady drizzle.
Once about half of the oil has been added, still with the motor running, drizzle in the lemon juice slowly.
Continue adding the remaining oil in a slow, steady stream while the processor runs, scraping down the sides as necessary; this emulsification process takes about 20 minutes and must be done slowly for a stable sauce.
With the motor running, drizzle in the ice water. The mixture will become smoother and very creamy; stop and scrape down the bowl if needed, then transfer the sauce to a bowl.
Optionally drizzle extra virgin olive oil on top and sprinkle with chopped parsley. Serve with flatbread or fresh vegetables.