Heat a few tablespoons of olive oil in a large skillet over medium-high heat. Fry eggplant slices in hot oil until browned thoroughly, 3 to 5 minutes per side.
Add oil between batches as needed.
Drain fried eggplant slices on paper towels and season with salt.
Heat a large skillet over medium-high heat. Cook and stir beef in the hot skillet until browned and crumbly, 5 to 7 minutes.
Add onion and garlic to the beef; cook and stir until onion is golden brown, about 5 minutes more.
Pour spaghetti sauce over the ground beef; add oregano and stir. Simmer mixture until sauce is heated through, 5 to 7 minutes.
Arrange about half the eggplant slices into the bottom of a shallow casserole dish to cover completely. Ladle about half the beef sauce over the eggplant.
Sprinkle 1/4 cup Parmesan cheese over the sauce. Repeat layers.
Beat egg with 2 tablespoons Parmesan cheese; pour over the top of the casserole.
Bake in preheated oven until bubbling around the edges and hot in the center, 20 to 25 minutes.