Step 1Heat the oven to 350 degrees. Using your fingers, spread a thin layer of olive oil over the bottom of a deep 13x9-inch baking pan and set it aside.
Step 2Fill a large pot with 4 or 5 quarts of water. Add 2 tablespoons of salt and bring the water to a boil. Add 1 lasagna strip at a time. Gently stir when they are all added so they don't stick to each other. Turn the heat down a little but keep the water boiling. Cook until the noodles are soft and pliable, 10 minutes. Drain the water and fill the pot with cold water. Use your fingers to move the noodles around so they don't stick to each other.
Step 3Slice the stem off the top of the eggplant and discard. Cut the eggplant down the center lengthwise. Cut each half eggplant lengthwise into 3 long slices, then slice the 3 long strips into strips 1 inch wide. Holding the strips together, cut them crosswise into 1-inch cubes. Spread the cubes out and lightly salt them. Set aside.
Step 4Cut the stem top and root bottom off the onion and peel off the papery skin. Slice the onion from the stem top down, then slice each half into about 4 or 5 slices. Keeping the slices of each half stacked, slice the stack crosswise into about 5 strips. Still holding the cut half onion, chop each half into 1/4-inch to 1/2-inch pieces. Spread the onion pieces out and lightly salt. Set aside.
Step 5Chop each garlic clove into small pieces and add to the onion. Cut the top core out of each tomato. Slice the tomatoes lengthwise through the center. Place the halves flat side down and cut them into small cubes. Spread them out and lightly salt. Set aside.
Step 6Put 2 tablespoons of the olive oil in a large skillet. Tilt the skillet so the oil coats the bottom. Turn the heat to medium-high and add the eggplant, onion, garlic and tomatoes. Cook, stirring frequently, until the vegetables are slightly cooked, about 5 minutes. (You may need to do this in 2 batches if your skillet is not large enough.) Add about 1 1/2 cups of the tomato sauce, stir to mix well with the vegetables, and cook just another couple of minutes. Remove from the heat.
Step 7Place 3 strips of lasagna side by side on the bottom of the baking dish. Spoon a layer of vegetables over the strips. Spread 1 cup of the cottage cheese over the vegetables. Put another 3 strips of lasagna over the first layer and repeat, adding vegetables and cottage cheese. Put the last 3 strips of lasagna over the second layer of filling and stir the remaining olive oil into the remaining 1/2 cup of tomato sauce. Spread this evenly over the lasagna.
Step 8Bake the lasagna until the eggplant is soft, 40 minutes. Remove from the oven. If using Parmesan cheese, sprinkle it evenly over the top and it will melt immediately. Cut the lasagna into large squares and serve hot.