In a 4- to 5-quart pan, combine oil, garlic, onion, carrot, and celery. Stir often over high heat until vegetables begin to brown, 6 to 8 minutes. Pour from pan into a bowl.
To pan, add lamb and broth; cover and bring to a boil on high heat. Reduce heat and boil gently for 10 minutes. Uncover, turn heat to high, and stir often until juices evaporate, drippings are very browned, and meat is lightly browned, 8 to 10 minutes. Return browned vegetables to pan.
Add tomatoes with their liquid, wine, fennel seed, and sage; stir to free browned bits. Bring to a boil, then reduce heat, cover, and simmer for 40 minutes; stir occasionally.
Meanwhile, rinse fresh artichokes, drain, and break off outer coarse leaves down to the very tender inner yellow ones (easily pierced with your fingernail). Cut off thorny artichoke tips, peel coarse fibers from bottoms, and cut artichokes in half lengthwise. In a bowl, mix with 2 cups water and vinegar; drain. If using frozen artichokes, omit water and vinegar.
Add artichokes and potatoes to lamb mixture. Cover and simmer until potatoes and lamb are very tender when pierced, about 30 minutes longer; stir occasionally. Pour into a bowl, sprinkle with parsley, and season to taste with salt and pepper.