Combine flour, salt and pepper in a shallow dish. Season lamb with salt and pepper as well, then dredge lamb pieces in flour and tap off excess.
Set a large, heavy-bottomed Dutch oven over medium-high heat. Add olive oil and wait until it smokes. Place lamb pieces in the pot. Brown on all sides, about 3 minutes per side. Remove to a plate and set aside.
Add garlic, carrot and onion to pot. When onion is soft and translucent, 3-5 minutes, add stock and wine. Add bay leaf, thyme, cloves and barley. Return the meat to the braising liquid. Cover and transfer to a preheated 350°F oven and cook 2 hours.
Arrange potatoes, cut side down, over the top of the stew, completely covering the surface (you may need to cut the last one or two to make it fit). Drizzle with oil and sprinkle with salt and pepper. Return to oven, uncovered, for an additional 30 minutes.
Meanwhile, melt butter in a small pot. Chop garlic roughly, then drizzle with olive oil and salt. Chop more finely, then press and mash with the side of your knife blade until it forms a paste. (Alternatively, combine garlic with oil and salt in a mortar and grind with pestle.) Add chopped parsley and add to the melted butter. Cook very slowly until heated all the way through.