Place the garlic and 3/4 cup water in a small saucepan. Cover and bring to a boil.
Reduce the heat to medium-low and simmer until the garlic is tender, about 15 minutes. Keep the lid around, as you don't want to evaporate off all of the water. You're looking to boil the garlic until it's fork-tender.
Combine the garlic, the water, and the next 4 ingredients in a blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Keep a clean towel over opening in blender lid while firing it up.
Once it's running, remove the towel, and slowly drizzle in the olive oil, allowing it to emulsify into the vinaigrette.
Serving ideas: Grilled snapper (add one roasted jalapeño, a squeeze of lime, and a handful of cilantro), sautéed shrimp (exchange a tablespoon of olive oil with one of melted butter and throw in a dozen or so leaves of basil), slow-roasted lamb (add juice of half a lemon and 1/2 teaspoon dry-pan-toasted cumin).