Heat about 1 inch extra-virgin olive oil in a deep skillet over medium heat. Pat eggplant slices dry with a paper towel and fry eggplant in batches in the hot oil until golden, 2 to 3 minutes per side. Transfer eggplant to a paper towel-lined plate using a slotted spoon; season with salt.
Combine tomato puree (passata), 1 tablespoon olive oil, garlic, salt, and 4 basil leaves in a saucepan; simmer over medium heat for 10 minutes. Remove garlic from sauce.
Preheat oven to 375 degrees F (190 degrees C). Lightly grease a baking dish.
Place a layer of fried eggplant in the baking dish and cover with a layer of tomato sauce. Top with a layer of mozzarella cheese, oregano, Parmesan cheese, and chopped basil leaves, in that order. Continue layering in the same order with remaining ingredients, ending with a layer of mozzarella cheese and a drizzle of olive oil.
Bake in the preheated oven until heated through and cheeses are melted, about 40 minutes. Let stand a few minutes before slicing.
