Preheat the oven to 400 degrees . Line 2 baking sheets with parchment paper and drizzle 1/2 tablespoon olive oil on each. Place half of the eggplant slices on each pan, turn to coat with the oil, then arrange in a single layer. Bake until tender and lightly browned, about 40 minutes.
Meanwhile, in a large, deep skillet, heat the remaining 1 1/2 tablespoons olive oil over medium heat; add the garlic and cook, stirring, until just golden, 1 to 2 minutes. Add the tomatoes; bring to a boil. Lower the heat and cook, partially covered and stirring occasionally, until the sauce is thickened, about 25 minutes. Season with salt and pepper.
In a large pot of boiling, salted water, cook the noodles, stirring often, until tender, about 5 minutes. Drain the noodles and rinse with cold water.
Line a 9-inch square baking dish with two 16-inch-long sheets of parchment paper, allowing a few inches of overhang on all sides. Using a food processor, puree the cottage cheese. Add the egg and 1/4 cup parmesan, season with salt and pepper and pulse to combine.
Spread 1 cup tomato sauce in the bottom of the prepared dish. Lay 2 noodles side by side over the sauce. Layer half of the cottage cheese mixture, half of the eggplant slices and one-third of the mozzarella on top. Spoon 1 1/4 cups tomato sauce over the mozzarella and top with 2 more noodles. Layer with the remaining cottage cheese mixture and eggplant, another third of the mozzarella, 1 1/4 cups tomato sauce and the remaining 2 noodles. Spread the remaining tomato sauce on top and fold in the parchment to cover the lasagna.
Cover the dish tightly with foil, place on a baking sheet and bake for 35 minutes. Remove the foil, sprinkle the remaining mozzarella and 2 tablespoons parmesan on top and bake until bubbly, about 10 minutes. Let cool for 10 minutes before serving.