Step 1Trim the root end and tops off the garlic, so that you have the white part and about 4 inches of the green. Discard any yellowed or old-looking leaves. Cut the garlic plants in half lengthwise and then cut crosswise into thin slices (you should have 5 to 6 cups).
Step 2In a wide pot, heat 3 tablespoons of oil, the garlic and onion over a medium-low heat. Add one-half to 1 teaspoon salt, stir and cook, stirring occasionally, until the garlic is wilted but still bright green, 5 to 10 minutes. Reduce the heat to low, cover and cook until the garlic and onion are tender, about 10 minutes more.
Step 3Add the potatoes, 2 cups water, a little salt and several grinds of pepper and cover the pot. Raise the heat and bring to a gentle boil. Reduce the heat to a simmer, cover the pot partially, and cook for 5 minutes. Add 3 cups water and continue cooking partially covered until the potatoes are tender and soup flavors develop, about 15 minutes.
Step 4Add 1 cup water if the soup seems thick. Season to taste with salt, pepper and 1 to 2 tablespoons oil. Drizzle each serving with additional oil if desired.