Preheat the oven to 500 degrees.
Prep the eggplants by discarding the tip and stem ends; then cut the eggplants into 1-inch cubes. Divide between two baking sheets. Drizzle each portion with 1/4 cup plus 2 tablespoons of the olive oil. Toss and gently massage the cubes to coat the eggplants. Spread into a single layer. One batch at a time, place the baking sheet on the oven floor or lowest rack and cook for 15 to 20 minutes, until the cubes are dark brown all over, tossing occasionally with a metal spatula. NOTE: this'll produce a lot of steam, so be careful when you open the oven door. Also: this might set off your smoke detector. Remove the baking sheet from the oven and season cubes with 1 teaspoon of the salt.
Combine the garlic and remaining 1/4 cup olive oil in a large Dutch oven. Cook until the garlic is soft and very light golden brown, stirring occasionally so it doesn't burn, about 5 minutes. Add the tomatoes, sugar, and the remaining 1 1/2 teaspoons salt, stir to combine, and cook the tomatoes for 5 to 6 minutes, until they thicken slightly. Spoon out and reserve 1/2 cup of the tomato sauce. Add the roasted eggplants, olives, za'atar, pepper, and red chile flakes. Stir to combine and cook until the eggplants have taken on the color of the sauce, about 5 minutes more. Add a splash of water (up to 3/4 cup) if the sauce is too dry and sticky. Taste and adjust for seasoning; then set aside.
Now it's time to make the cheese sauce! Combine the butter, onion, chile pod, and bay leaves in a medium heavy-bottomed saucepan over medium-low heat. Cook, stirring often, until the butter is melted and the onion is soft but not brown, about 10 minutes, stirring often to prevent the onion from browning. Add the flour, whisking constantly, and cook for 2 minutes. Gradually add 1 cup of milk, whisking constantly. Increase heat to high and cook the sauce, whisking constantly, until it thickens. Add another cup of milk, continue to whisk constantly. Return the sauce to a boil and add the remaining 2 cups milk, whisking constantly. Return the sauce to a boil, reduce the heat to low, and simmer, whisking, until the sauce is thick enough to coat the back of a spoon -- 3 to 4 minutes.
Turn off the heat and stir in the salt. Pour the sauce through a fine-mesh strainer into a medium bowl and discard the contents of the strainer. Ladle out 1 cup of the white sauce (this'll go on top of the lasagna later) and set it aside. Add the mozzarella, ricotta, Parmesan, and eggs to the bowl with the remaining white sauce and gently whisk to integrate the additions into the sauce.
Time to make the lasagna. Exhausted yet? Preheat the oven to 350 F.
Combine 6 quarts water and 6 tablespoons salt in a large soup or pasta pot and bring to a boil over high heat. Prepare an ice bath in large bowl and create a bed of clean dish towels.
Drop the lasagna into the boiling water, stir, and boil the pasta one minute less than it says to on the package. Drain the pasta (or use tongs to lift it out) then add directly to the bowl of ice water. When cool, lay on the dry kitchen towels to dry. (If you're laying them on top of one another, use olive oil so they don't stick.)
To assemble your lasagna: spread 1/2 cup reserved tomato sauce over the bottom of a 13 X 9-inch baking dish. Lay 3 sheets of lasagna on the bottom of the pan to cover it. Add half of the tomato eggplant mixture (about 2 1/2 cups), and spread it evenly over the pasta. Spoon half of the cheese sauce over the sauce and use another clean spoon or spatula to push it to the edges. Lay 3 sheets lasagna on top. Press gently down. Spoon the remaining tomato and eggplant sauce on top of the pasta. Spoon the remaining cheese sauce over the tomato and eggplant sauce. Lay a final layer of pasta on top of the cheese sauce.
Using a rubber spatula, gently spread the reserved bechamel (white sauce) evenly over the top of the lasagna and sprinkle the Parmesan over the bechamel. Using a long sharp knife, cut the lasagna into 9 or 12 equal-size servings (this makes it much easier to serve later).
Place the lasagna on a baking sheet and bake in the oven for 35 to 45 minutes, until the edges are golden and puffed up.
Turn on the broiler and carefully broil the top of the lasagna until the top is golden brown all over. Keep an eye on it, so it doesn't burn.
Remove the lasagna from the oven and let it cool for at least 1 hour before serving (!!!). (That's Nancy's hardest step. See if you can pull it off. I think it helps the lasagna set.) Enjoy!