Heat oil in a large Dutch oven over medium-high heat. Sprinkle lamb with salt and pepper, add to the pot and cook until browned on all sides, about 10 minutes.
Transfer lamb to a plate and add onions to the pot. Cook, stirring occasionally, until onions are golden brown, about 10 minutes. Add wine and simmer until reduced by about half. Add broth, return lamb to the pot, cover and simmer two hours.
Add apricots, turn lamb and continue to simmer until lamb is very tender, about 1 hour more.
Transfer lamb to a platter, cover and keep warm.
Simmer sauce until reduced to about 2-1/2 cups; taste and adjust seasoning with salt or pepper if desired.
Slice lamb, pour sauce over lamb and serve.
Long, slow cooking makes lamb meltingly tender. Garnish the dish with chopped fresh dill or thyme, if you like. Serve with mashed potatoes, polenta or egg noodles for soaking up the sauce. Serves 6 to 8.