After peeling the cloves, I always split them in half lengthwise and remove the germ, as the little sprout in the center can leave a noticeable sharpness in raw applications. (Read my article on removing the garlic germ for a more in-depth explanation.)
I process the cloves in a food processor, along with kosher salt for some added friction. Initially, the chopped garlic will want to spray along the sides of the bowl, so you have to frequently stop to scrape down the sides, pulsing until the garlic is completely puréed. The garlic needs to be fully broken down in order for the proteins and stabilizers to be released from within its cell walls.
Once the garlic is smooth, I blend in only some of the fresh lemon juice. With mayo, all of the liquid can be added at the start because the egg is such a strong emulsifier. But because garlic is a weaker emulsifier, we need all the help we can get, and keeping the paste thick creates more drag on the oil droplets, helping to keep them apart.