In a food processor or blender, combine the garlic, anchovies, and a pinch of salt and puree until smooth, about 15 seconds. Add the egg yolks (or mayonnaise), vinegar, and lemon juice and puree again until smooth, another 15 seconds.
With the machine running, stream in the oil very slowly until emulsified. (Note: If you have a Cuisinart, stream the oil through the food pusher insert of the food chute… there is a teensy hole at the bottom that will allow the oil to stream in very slowly.) Taste and adjust with more salt and/or lemon juice. If necessary, with the machine running, thin the dressing with 3 to 4 tablespoons of water.
Transfer the dressing to a jar and store in the fridge for up to 2 weeks.
