Preheat an oven to 450°F. Evenly space out 2 oven racks.
Lay the eggplant in an even layer on two or more rimmed baking sheets. Brush both sides of the eggplant slices with oil, then season each side with oregano, salt and pepper. Bake for 15 minutes, then flip and bake until other sides are lightly browned (about 10 minutes more). Set aside. Reduce oven temperature to 400°F.
Meanwhile, bring a large pot of salted water to a boil. Cook lasagna sheets for 4 minutes. They will still be fairly firm. Drain and rinse under cold water. Transfer cooked lasagna sheets to a parchment-lined baking sheet in even layers separated by parchment.
Add ricotta cheese, 3 ounces mozzarella cheese, 1 1/2 ounces grated Parmesan and chopped basil to a medium bowl, mixing to combine. Season to taste with salt and pepper.
Spread 3/4 cup of tomato sauce in an even layer in the bottom of a 9 x 13 baking dish. Top with 1/4 of the cooked lasagna sheets, arranged in an even layer. Top the noodles evenly with 1/3 of the eggplant, then 1/3 of the cheese mixture. Repeat even layers two more times with remaining sauce, lasagna sheets, eggplant and cheese mixture. Finish with remaining 3/4 cup of tomato sauce and lasagna sheets. Top evenly with remaining 4 ounces mozzarella and 1 1/2 ounces Parmesan cheese.
Cover with foil and bake for 40 minutes. Remove the foil, then bake until the cheese is melty and golden on top (about 15 minutes more). Let cool for 10 minutes before serving.