In a large Dutch Oven, heat the olive oil till smoking and in it brown the pieces of lamb (you may have to do this in 2-3 batches) until golden-brown all over.
When all the pieces are browned add onion, garlic, water and wine (you could use just water if you wanted to), salt and pepper to taste, and simmer for 1 1/2 hours, or until meat is very tender.
Add lettuce and dill. You may have to cram it into the pot but it will reduce to practically nothing. Put the lid on the Dutch Oven and cook for 8 minutes. Remove lid, stir carefully not to break up meat and cook another 2 minutes. Turn off heat.
In a large, heatproof bowl, whisk together eggs, cornstarch, lemon juice and a little salt and pepper to taste (remember, there is more in the pot).
With a ladle, remove ladlefuls of the cooking liquid the meat and lettuce have cooked in and VERY slowly whisk it into the egg/lemon mixture. Keep doing this until, whisking all the while, until you have incorporated most of your cooking liquid - this tempers the egg-lemon sauce so it won't curdle when added to your lamb.
Pour all the sauce into the pot, covering the lamb and lettuce. Turn heat up to medium-high and just bring to the boil. Cook for 1-2 minutes.
Dinner is ready. Have lots of crusty bread around to sop up the delicious avgolemono sauce.