For the cannelloni: Fill a large pasta pot with an insert with water. Bring to a boil. Season with salt and cook the lasagna sheets for about 1 to 2 minutes. Fill a large bowl with water and ice. Drain the lasagna sheets and place them in the ice bath. When cool, drain the pasta again, spray the sheets with nonstick spray to keep from sticking and lay them on a sheet pan. Set aside.
Heat the oil in a large skillet over medium heat. Add the onion and garlic and cook about 1 minute. Season with salt and pepper. Add the eggplant and cook until softened, about 5 minutes. Transfer to a large bowl. Add the oregano and smash the eggplant mixture with a potato masher. Stir in the panko, 1/2 cup of the Parmigiano and the egg. Set aside.
For the tomato sauce: Heat the oil in a deep medium skillet over medium heat. Add the shallots and garlic and cook about 5 minutes. Stir in tomato puree and basil and season with salt and pepper. Bring to a simmer, adjust heat, and simmer 10 minutes.
For the anchovy breadcrumbs: Heat the oil and butter in a medium skillet over medium heat. Once bubbling, add the anchovies. Cook, stirring, until the anchovies melt, about 1 minute. Add the panko and toss until coated. Cook until the panko is golden, stirring occasionally, about 2 minutes. Remove from the heat, stir in the parsley and season with salt and pepper. Set aside.
Ladle 1/2 cup of the tomato sauce into an 8-inch-square baking dish, creating a bed for the pasta.
Put about 1/4 cup of eggplant filling on a lasagna sheet. Starting with the shortest end, roll it up like a cigar away from you. Continue until all the pasta and filling is used. Lay each cannelloni into the baking dish. Top with more tomato sauce and remaining 1/4 cup freshly grated Parmigiano. Bake until brown and bubbly, about 10 minutes. Remove from the oven and sprinkle with more Parmigiano and with the anchovy breadcrumbs.