Take your cold water and add the salt. When it dissolves, add the cubed eggplant. Cover it with plastic wrap and let it sit for 20 minutes or so.
While the eggplant gets a soak, heat your skillet over medium high. Add a tablespoon of oil and then brown the ground lamb. Set it aside in a bowl after draining the juices.
Drain the water from the eggplant. Dry it and blot it on paper towels.
Heat your skillet back to medium-high. Add 1/4 cup of olive oil and the eggplant. Cook until soft (about 5 min.) and then remove with a slotted spoon to the bowl with the lamb.
Toss the rest of the oil into a the skillet (still on medium-high) and cook the onion and garlic for about 3 minutes. Stir in the eggplant and lamb. Add the wine, tomato sauce, parsley, oregano, and cinnamon.Lower heat and simmer, with occasional stirs for around 15 minutes. Add pepper and salt as needed.
In a bowl, beat the eggs. In another skillet, over medium-high, melt the butter and then lower the heat a bit while you stir in the flour. Create a paste. Whisk in the milk, slowly. Bring it to a boil (over low heat) and then remove from the heat.
Add half of the the hot mix into the eggs and stir well, Then pour that bowl into the skillet you have the mix in. Add cheese and dill. Give it a few more stirs.
In the main skillet, smooth the lamb and then slowly pour the topping mixture from the other skillet over it.
Bake for 35-45 minutes until the top seems to set and brown. Remove and let sit for 5 minutes.