Rinse and dry the lemons. With a sharp vegetable peeler or paring knife, shave off the outer yellow zest in strips, taking none of the bitter white pith, until you have about ¼ cup of strips, packed together. Juice some or all of the lemons, straining to remove seeds and pulp, to get about 1/3 cup of lemon juice.

Put the juice, strips of peel, garlic, onion, and salt in a 2 quart saucepan; pour in the water and 1 tablespoon of the oil. Bring to a boil, cover, and cook for about 20 minutes at a gentle boil. Remove the cover and cook rapidly for 20 or 30 minutes, or until the overall volume has reduced by half and the onion pieces are barely covered in liquid.

Puree the warm sauce in a blender or food processor. With the machine running, pour the remaining tablespoon of olive oil in a thin stream to incorporate it into the sauce.

Taste the sauce and blend in salt if needed; reheat if necessary before serving. Store in the refrigerator.