Peel the eggplant and slice it lengthwise into 1/4-inch slices. Sprinkle each layer with salt and place into a colander, overlapping and salting as you go. After you fill the colander, place a plate on top and weigh it down with a heavy pan or a tea kettle filled with water. Let the eggplant sweat for at least 30 minutes.
While the eggplant sweats, make the tomato sauce: Cover the bottom of a sauce pan with olive oil and heat over medium-high heat. Add the sliced garlic and let it cook until it sizzles (do not brown the garlic, as it will quickly become bitter). Add the tomatoes with their juices and a big pinch of salt. Coarsely crush the tomatoes using a potato masher or pair of scissors. Lower the heat and simmer until the tomatoes are reduced by almost half, stirring frequently.
Heat the oven to 450°F. Cover the bottom of two sheet pans with olive oil.
While the oven heats up, remove the eggplant from the colander and thoroughly pat dry each slice. Dredge the eggplant slices in flour, shaking off any excess. Place on the sheet pans and drizzle each slice with olive oil. Bake until browned on one side (about 15 minutes), then turn over and brown the other side. Repeat until you have cooked all the eggplant.
Using a 7x11-inch baking dish spread a thin layer of sauce on the bottom and layer the eggplant until it completely covers the bottom.
Sprinkle the eggplant generously with the grated Parmigiano-Reggiano. Add another layer of sauce and then the eggplant. Continue to build the layers until you are about two layers from the top, then add a single layer of sliced mozzarella. Finish with a couple more layers of eggplant, sauce, and Parmigiano-Reggiano. Finish the top with more Parmigiano-Reggiano.
Lower the oven temperature to 400°F. Bake on the upper third rack for 20 minutes. Check in: If the dish looks especially liquid, remove it from the oven and discard a few spoonfuls. Bake for another 15 minutes or so. Let stand for 15 to 20 minutes before serving.