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 olive oil in a large skillet over medium-high heat. 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 of the potatoes, and 1/2 the feta into a 13x9-inch baking dish. Pour 1/2 of the 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 for sauce 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 top with Parmesan cheese. Bake, uncovered, for another 25 to 30 minutes.