With a paring knife, make small incisions in leg and insert slivers of garlic. Rub meat with olive oil. Salt and pepper to taste.
Layer the herbs and vegetables (except potatoes) in bottom of heavy roasting pan with tight-fitting lid. Lay goat on top and pour in wine and broth. Place on stove and bring liquid to a boil.
Cover and place in oven. After 30 minutes, turn heat down to 300 degrees. Braise until meat is so tender it's almost falling off the bone. This will take 5 to 7 hours, depending on size of leg, roasting pan and oven. Check from time to time, basting with liquid and adding more if liquid evaporates too much. With about an hour to go before the goat is done, bury potatoes around meat.