Using a mortar and pestle, pound and then press garlic and sea salt into a smooth paste. If you don't have a mortar and pestle, use the flat of a knife to smush and spread minced garlic on a cutting board until smooth.
Still whisking vigorously, add olive oil a very small drizzle at a time, whisking until each addition is fully combined. Continue adding just a drizzle, until you've added about a half cup of oil. From there, add oil in more generous glugs, making sure all additions are fully emulsified before adding any more. If mixture becomes too thick, whisk in a half tablespoon water – just enough to loosen the mixture slightly. Add more water if needed, but proceed with caution.