Step 1
Sprinkle eggplant slices with salt and set aside for 30 minutes. Rinse and pat dry.
Step 2
Preheat oven to 375 degrees F (190 degrees C).
Step 3
Heat oil in a large skillet over medium-high heat. Rinse and pat eggplant dry. Cook eggplant and zucchini in hot oil until lightly browned on both sides about 3 minutes per side; remove with a slotted spoon to drain on a paper towel-lined plate reserving as much oil as possible in the skillet
Step 4
Adding more oil to skillet as needed and let it get hot. Cook potato slices in hot oil until browned 3 to 5 minutes per side; remove with slotted spoon and drain on a paper towel-lined plate again reserving oil in the skillet.
Step 5
Saute onion and garlic in reserved oil until lightly browned 5 to 7 minutes. Pour in vinegar bring to a boil and reduce heat to medium-low; cook until liquid is reduced in volume and thick. Stir in tomatoes lentils 1/2 the juice from lentils oregano and parsley. Cover reduce heat to medium-low and simmer 15 minutes.
Step 6
Layer about 1/3 of the eggplant 1/3 of the zucchini 1/2 the potatoes 1/2 the onions and 1/2 the feta into a 13x9-inch baking dish. Pour tomato mixture over vegetables; repeat layering finishing with a layer of eggplant and zucchini.
Step 7
Cover and bake in preheated oven for 25 minutes.
Step 8
Stir butter flour and milk together in a small saucepan; bring to a slow boil whisking constantly until thick and smooth. Season with pepper and nutmeg; stir. Remove from heat cool for 5 minutes and stir in beaten egg.
Step 9
Pour sauce over vegetables and sprinkle with Parmesan cheese. Bake uncovered for another 25 to 30 minutes.