Peel the eggplant and cut the flesh into 1-inch cubes, more or less. Drop the cubes into boiling salted water and cook about 5 minutes, just until cooked. Drain well.
Meanwhile, slice the mushrooms. There should be about 2cups. Heat 1tablespoon butter in a skillet and add the mushroom slices. Sprinkle with salt and about 1teaspoon lemon juice. Cook, stirring and tossing, until the mushrooms give up their juice. Continue cooking until the liquid evaporates. Set aside.
Melt 1½tablespoons butter in a saucepan and add the flour, stirring with a wire whisk. Add the milk and cream, stirring rapidly with the whisk. When blended and smooth, add salt and pepper to taste, the remaining lemon juice, nutmeg, and Tabasco to taste. Stir in the mushrooms and eggplant. Stir in the egg. Spoon the mixture into a baking dish (an 8-inch pie plate works well). Sprinkle with a mixture of crumbs and cheese and dot with the remaining 1tablespoon butter. Bake30 to 40 minutes, and then brown under the broiler.