Cut off top of artichoke and gently pull open center. Trim bottom of stem (if present), keeping stem attached, and peel stem. Put artichoke in a large bowl of cold water. Repeat with remaining artichokes.
Remove artichokes from water. Heat 1/8 cup olive oil in pot over moderate heat until hot but not smoking, then add artichokes, thyme sprigs, garlic, 1/2 cup water, pepper, and 1/4 teaspoon salt. Cover pot and braise artichokes, turning occasionally, until artichokes are browned in spots and bases are tender when pierced with a knife, about 35 minutes.
Transfer artichokes, thyme, and garlic to a platter. Add up to 1/2 cup water to pot and deglaze by boiling over high heat, stirring and scraping up brown bits, 1 minute. Pour pan juices (they will be dark) into a small bowl and stir in remaining 1/2 teaspoon salt. Squeeze pulp from 2 of garlic cloves into juices and mash into sauce with a fork.
Divide artichokes and remaining garlic cloves among 6 plates and drizzle with sauce. Garlic cloves can be peeled and spread on crusty bread.
adapted from Gourmet, April 2005, by Maggie Ruggiero