Heat the oven to 450 degrees F. Turn off the oven and put the bread directly on the oven rack. Let the bread dry out for at least 8 hours or up to overnight in the turned-off oven (I didn’t really plan ahead enough to do this, so I just lightly toasted the bread in my toaster oven to just dry it out). To reduce the bread to coarse crumbs, pass it through a food mill fitted with the large die, or, using short pulses, process the bread in a food processor
Heat 2 tablespoons of the oil in a large saute pan over medium heat. Add the bread crumbs and toast them, stirring or shaking the pan a few times, until golden brown, 4 to 6 minutes. Set aside.
Heat the oven to 400 degrees F. Cut the eggplants in half lengthwise and score each half in a large crosshatch pattern with a sharp knife. Set the eggplants, cut side up, on a rimmed baking sheet and season the flesh with salt and pepper. Strip the leaves from the thyme sprigs onto the eggplant, then drizzle the eggplant evenly with the remaining 4 tablespoons olive oil.
Roast the eggplants until very tender and golden brown, 30 to 35 minutes. Let the eggplants cool slightly, then scoop the flesh from the skins into a food processor. Using short pulses, coarsely puree the flesh. Reserve the puree.
Bring a large pot of salted water to a boil. Drop in the entire pasta sheet and blanch it for 30 seconds, stirring and separating it with tongs to keep the sheet from sticking to itself. Carefully remove the pasta sheet and lay it out on kitchen towels or paper towels. Trim the edges square (honestly, I didn’t feel the need to trim mine since it was fairly squared off already when I rolled it).
Spread the eggplant puree evenly over the entire sheet of pasta, all the way to the edges (you may have a bit left over). Top the eggplant evenly with the mozzarella and 1/4 cup (25 grams) of the Parmesan, and then sprinkle with salt and pepper. Starting from a short side, tightly roll up the sheet. The rotolo should look like a jelly roll.
Heat the oven to 500 degrees F. Spread the tomato sauce in the bottom of a shallow 2-quart baking dish, or for individual servings, spread 2 tablespoons of the sauce in the bottom of each of four 2-cup baking dishes. Using a sharp plain-edged or serrated knife, trim the ends of the rotolo (again, I think this is unnecessary since you can easily put the “ugly” edges on the bottom and no one will even notice the difference), then cut it crosswise into four slices, each about 1 inch thick. Lay the slices flat in the baking dish(es) and sprinkle the bread crumbs and the remaining 1/4 cup (25 grams) Parmesan evenly over the slices.
Bake the rotolo until heated through and lightly browned on top, 10 to 15 minutes.
Transfer the rotolo slices to warmed plates if baking in a single dish, or leave in the individual baking dishes. Garnish each serving with some of the oregano, if desired.