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 35 F oven for 1 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 5 minutes longer or until golden brown.
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