Preheat the oven to 375°F (190°C).
Heat a large skillet over medium-high heat. Add the ground beef or turkey and cook until browned, breaking it up with a wooden spoon as it cooks. Drain off any excess fat.
Add the tomato sauce to the skillet with the browned meat and stir to combine. Season with salt, black pepper, and dried oregano. Reduce the heat to low and let simmer for 10 minutes.
While the meat sauce simmers, lightly grease a 9x13-inch baking dish with olive oil.
Layer half of the eggplant slices in the bottom of the baking dish, overlapping slightly if necessary. Sprinkle with half of the parmesan cheese and half of the chopped basil.
Spoon half of the meat sauce on top of the eggplant layer and spread it evenly.
Repeat with the remaining eggplant slices, parmesan cheese, basil, and meat sauce.
In a mixing bowl, whisk together the beaten eggs and shredded mozzarella cheese. Pour the mixture over the top of the lasagna, spreading it evenly with a spatula.
Cover the baking dish with foil and bake in the oven for 30 minutes.
Remove the foil and bake for an additional 15 minutes, until the top is golden brown and the cheese is melted and bubbly.
Remove from the oven and let cool for a few minutes before slicing and serving.