Preheat oven to 375 degrees. Place noodles in an 8-by-8-inch baking dish, and cover with hot tap water; set aside to soften.
In a 5-quart Dutch oven or heavy pot, heat oil over medium-high. Add onion, eggplant, and garlic; season with salt and pepper. Cover and cook, stirring occasionally, until eggplant is very tender, 8 to 10 minutes. Add sirloin, and cook, breaking up meat with a spoon, until no longer pink, 3 to 5 minutes. Add tomato puree, and cook until thickened, 3 to 5 minutes. Season meat sauce with salt and pepper.
Meanwhile, in a medium bowl, combine cottage cheese and 1/4 cup Parmesan; season with salt and pepper. Remove noodles from baking dish, discarding water.
Spread about 1/4 cup meat sauce in bottom of dish, and top with 4 noodle halves. Layer with 1/3 cheese mixture, then 1/3 sauce. Repeat twice with remaining noodles, cheese mixture, and sauce. Sprinkle with mozzarella and remaining 2 tablespoons Parmesan. Bake until lasagna is bubbling and cheese topping is golden, 30 to 35 minutes. Let stand 10 minutes before cutting and serving.