Make the sauce: In a blender, pulse the toma­toes and their juices until just slightly chunky.In a medium–large (4-quart or so) heavy-bottomed pot over medium-low heat, add the oil and garlic, and cook, stirring occasion­ally, until the garlic is softened slightly and fragrant but not yet browned, about 4 min­utes. Pour in the puréed tomato mixture and season with a generous pinch each of salt and pepper. Bring to a gentle simmer and cook, stirring frequently, until thickened slightly, at least 1 to 1.5 hours. The sauce can be cooled and refrigerated for up to 2 days. If you’re baking and serving right away, set a rack in the top third of the oven and pre­heat to 400°F. If not, skip the preheat for now. Bread the eggplant: In a medium baking dish or wide shallow bowl, beat the eggs with 1 tablespoon water. Season with a generous pinch of salt. In a second baking dish or bowl, stir together the bread crumbs, Italian seasoning, and 1/2 teaspoon salt. Working with one or two at a time, dredge the eggplant slices in the egg wash and let the excess drip back into the bowl. Transfer to the bread crumbs and coat very lightly on each side. Line a large baking sheet with a few layers of paper towels and set by the stove. In a large high-sided skillet over medium-high heat, heat ½ cup oil until shimmering. Turn the heat down to medium and add some eggplant slices in a single layer until the skillet is full. Cook, turning once, until well browned on each side, about 6 minutes total. Transfer to the pre­pared baking sheet. Repeat with the remaining eggplant, working in batches as needed and adding a little more oil every other batch, or as needed. As you work, continue to add layers of paper towels between the eggplant slices so they remain separate. The fried eggplant can be refrigerated in airtight container for up to 1 day. In a 9-by-13-inch baking dish, ladle 1/2 cup plus 2 tablespoons sauce into the bottom. Cover the sauce with a single layer of eggplant (start with the thickest ones on the bottom and save the prettiest slices for the top). Ladle another 1/2 cup sauce on top, spread­ing it evenly. Sprinkle with about 1/3 cup plus 2 tablespoons mozzarella and 1 tablespoon Pecorino. Add another layer of eggplant and repeat this process until you’ve reached the final layer of eggplant. Top this layer only with 1/2 cup sauce, a final thin layer of mozzarella, and some Pecorino. (Reserve any remaining sauce for another use.) At this point, the eggplant parmigiana can be refrigerated, covered, for up to 1 day. Bake, uncovered, until the cheese is melted and bronzed in places and the sauce is bubbling around the edges, about 20 min­utes (add about 10 minutes if your dish was previously assembled and chilled). Remove and let cool slightly. Slice into squares and serve warm.