Sprinkle eggplant slices with salt and set aside for 30 minutes. Rinse and pat dry.
Preheat oven to 375 degrees F (190 degrees C).
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
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.
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.
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.
Cover and bake in preheated oven for 25 minutes.
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.
Pour sauce over vegetables and sprinkle with Parmesan cheese. Bake, uncovered, for another 25 to 30 minutes.
