Fill a large pot with water and bring to a boil over high heat. Add the lamb offal, lemon slices, bay leaves, salt, and peppercorns. Boil for 10-15 minutes, skimming off any foam that forms. Drain and discard the water.

Return the same pot to high heat and add the olive oil. Chop the onion and garlic into small pieces and add them to the pot. Cut the lamb offal into small pieces, add to the pot, and sauté for 8-10 minutes. Pour in the white wine, add the bouillon cube and water, cover, and simmer over medium heat for 30-40 minutes.

Slice the green onions, and roughly chop the lettuce and dill. Add them to the pot along with the rice. Cover and simmer over medium heat for 15-20 minutes.

For the avgolemono: In a bowl, combine the lemon zest and juice with the eggs. Whisk together. Slowly add 5-6 ladlefuls of hot soup broth to the egg mixture, whisking constantly.

Pour the egg mixture back into the pot, stir gently for a few seconds while shaking the pot, then remove from heat. Add the dill and stir. Serve with olive oil, dill, and freshly ground black pepper.