Slice eggplant, sprinkle with salt, and place in colander. Weigh down with a heavy plate for several hours. Then brush slices with oil and broil lightly. Saute beef and onions in 5 tablespoons of the butter. Add tomato paste, cinnamon, and salt and pepper and mix well. Sprinkle bottom of a greased baking dish with bread crumbs. Alternate layers of vegetables and meat in the pan, sprinkling each layer with cheese. The top layer should be vegetables. Dot with 1 tablespoon butter and add water. Set aside. In saucepan melt remaining 6 tablespoons butter. Add flour and cook, stirring, until well mixed and beginning to brown. Slowly stir in milk and cook, stirring, until sauce is smooth and thickened. Beat eggs with a little of the hot sauce, then stir in to remaining sauce. Remove from heat. Pour a little more than half the sauce over the vegetables and bake in a 350 F oven for 10 minutes. Meanwhile return remaining sauce to low heat and cook, stirring, until thick. Pour into casserole and sprinkle with cheese. Continue to bake for 45 to 50 minutes longer or until golden brown.