Slice ends off eggplants. Cut eggplants lengthwise into 1/4-inch-thick slices, discarding peel-covered ends. You should get roughly 16 slices total. Lay slices on a rimmed baking sheet and sprinkle both sides liberally with salt. Let stand for 15 minutes, then rinse salt off under cold running water and pat slices dry.
Preheat oven to 400ºF. Brush both sides of eggplant slices with olive oil and place in single layers on 2 baking sheets. Roast for 15 minutes, until tender, turning eggplant slices over halfway through. Let cool on sheets on wire racks until cool enough to handle.
In a large bowl, combine spinach, ricotta, garlic, eggs, 1/2 cup mozzarella and 1/2 cup Parmesan. Season with 1 tsp. salt and 1/2 tsp. pepper. Mist a 9-by-13-inch baking dish with cooking spray. Spread 1/2 cup of sauce over bottom of dish. Divide ricotta mixture among eggplant slices, using about 1/3 cup for each, spreading it down the center. Roll up slices and place seam-side down in baking dish. Top with remaining sauce and sprinkle with remaining mozzarella and Parmesan.
Cover baking dish with foil and bake for 30 minutes. Remove foil and bake until browned and bubbling, about 15 minutes longer. Let cool for 10 minutes before serving.