Position oven racks in the top and bottom thirds of the oven and preheat to 400 degrees F. Line 2 large baking sheets with parchment.
Cut off the stem of one eggplant, then slice 1/2 inch off one long side. Turn the eggplant onto the trimmed flat side and slice 1/2 inch off the 2 long sides, squaring it off slightly. Slice the eggplant into 6 long pieces, each about 1/4 inch thick. Repeat with the other eggplant. (Discard the trimmed pieces or reserve for another use.)
Divide the eggplant slices between the prepared baking sheets. Brush both sides of each slice with the olive oil and sprinkle generously with salt and pepper. Bake, flipping the slices over about halfway through, until just tender and starting to brown, 15 to 20 minutes. Use a large spatula to transfer the slices to a cutting board in a single layer to cool slightly.
Meanwhile, stir together the ricotta, parsley, lemon zest, basil, 2 cups of the mozzarella, 1/2 cup of the Parmesan, 1/4 teaspoon salt and a pinch of black pepper in a medium bowl. Taste and adjust the seasoning with salt and pepper if needed. Stir in the egg. Spoon a leveled 1/4 cup of the cheese mixture into the center of each eggplant slice and roll up tightly.
Spread half of the tomato basil sauce on the bottom of a 9-by-13-inch baking dish. Arrange the rolled eggplant slices seam-side down in the center of the baking dish, making 2 rows of 6 and leaving space between the rows. Spoon the remaining sauce down the middle of the rolls, leaving the edges exposed. Top the rolls with the remaining mozzarella and Parmesan.
Bake, rotating the dish halfway through, until the eggplant is very tender and the cheese is golden brown and bubbly, 40 to 45 minutes. Allow to sit for 10 minutes, then sprinkle with chopped parsley before serving.
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