How to Make It
Fill a large bowl with water and squeeze the lemon halves into it. Working with 1 artichoke at a time, trim the base, removing the stem and trimming the tough outer leaves beneath the heart. Cut off the top two-thirds of the leaves, leaving only the heart. With a small spoon, scrape out the hairy choke. Submerge the trimmed artichoke heart in the lemon water. Repeat with the remaining 5 artichokes. Once all the hearts have been trimmed, slice them 1/4-inch thick and return them to the lemon water.
In a large, deep saucepan, heat the 1/4 cup of olive oil. Add the garlic and cook over low heat until softened but not browned, about 5 minutes. Drain the artichokes and add them to the saucepan along with the potatoes, the 6 1/2 cups of water and a pinch of salt. Bring to a boil. Put the peppercorns in a tea ball or tie them up in a piece of cheesecloth and immerse them in the soup. Simmer until both the artichokes and potatoes are tender, about 30 minutes. Discard the peppercorns.
Using a fork or potato masher, partially mash the potatoes to thicken the broth. Season the soup with salt. Ladle into bowls and drizzle with olive oil. Serve with the lemon wedges.