Sprinkle eggplant slices with salt; set aside for 30 minutes. Rinse and pat dry with paper towels.

Preheat the oven to 375 degrees F (190 degrees C).

Heat olive oil in a large skillet over medium-high heat. Cook eggplant and zucchini in hot oil until lightly browned, 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.

Add more oil to the skillet if needed and let it get hot. Cook potato slices in hot oil until browned, 3 to 5 minutes per side. Remove with a slotted spoon to drain on a paper towel-lined plate, again reserving oil in the skillet.

Sauté 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 of the juice from lentils, parsley, and oregano. Season with salt and black pepper. Cover, reduce heat to medium-low, and simmer for 15 minutes.

Layer about 1/3 of the eggplant, 1/3 of the zucchini, 1/2 of the potatoes, and 1/2 of the feta in a 9x13-inch baking dish. Pour 1/2 of the tomato mixture over vegetables; repeat layering, finishing with a layer of eggplant and zucchini.

Cover and bake in the preheated oven for 25 minutes.

Meanwhile, make the sauce: Stir milk, flour, and butter for together in a small saucepan; bring to a low boil, whisking constantly, until thick and smooth. Season with nutmeg and black pepper. Remove from heat, cool for 5 minutes, and stir in beaten egg.

Pour sauce over vegetables and top with Parmesan cheese. Continue baking, uncovered, until sauce is bubbly and top is lightly brown, 25 to 30 minutes.