Place the eggplant cubes in a large colander. Sprinkle liberally with salt. Allow to stand and drain at least 30 minutes. Heat the olive oil in a large skillet over medium high heat. Add the eggplant. Cook, stirring frequently, until eggplant is tender and just beginning to turn golden brown, 10-12 minutes. Add the garlic and cook 1 minute longer. Turn off heat and set aside.

Bring a large pot of water to a rolling boil over high heat. Add the tablespoon of salt and the lasagna noodles. Cook, stirring occasionally, until the pasta is tender but firm, 5-7 minutes. Drain, rinse pasta under cold water. Separate the noodles so they don't stick together.

Preheat the oven to 350?. Prepare the Bechamel sauce and spoon a few tablespoons into the bottom of a 9x13 inch baking dish. Cover the bottom of the dish with a layer of lasagna noodles. They can overlap, but shouldn't come up the sides of the dish. Trim them with a knife to fit. Distribute half the eggplant evenly over the pasta. Pour 1 cup of the spaghetti sauce over the eggplant and use a rubber spatula to spread it evenly. Pour 1/2 cup of the bechamel sauce over the spaghetti sauce so it is distributed evenly. Cover with a layer of the grated mozzarella (about a third) and a sprinkling of parmesan. Repeat with another layer of lasagna noodles, the remaining eggplant, 1 cup of spaghetti sauce, 1/2 cup bechamel sauce, a third of the mozzarella, and a sprinkling of parmesan. Finish the lasagna with a layer of noodles and and the remaining mozzarella and parmesan. Cover the dish securely with aluminum foil. Bake on the top oven rack for 45 minutes. Remove the foil and continue baking until the top is just beginning to brown, about 15 minutes longer. Allow to stand 5 minutes before serving.

Bechamel Sauce:

Scald the milk, being careful not to boil it. In a small saucepan, heat the butter over medium-high heat until melted and bubbling. Add the red onion. Cook, stirring, until onion begins to soften. Add the flour and salt. Using a small wire whisk, stir to combine the butter and onion. Cook for 1-2 minutes. Turn off heat. Begin to add the milk to the butter and flour mixture a few tablespoons at a time, stirring well with the whisk after each addition. The mixture will seize up and become very thick at first, but as you add more milk and stir the mixture it regains is liquidity. After you have added about a cup of milk, the mixture will be fluid. Add all the remaining milk and stir well with the whisk. Turn the heat to medium. Cook, stirring, until the mixture is the consistency of heavy cream. Remove from the heat.