Heat broiler. Coat eggplant slices on both sides with cooking spray and season with salt and black pepper. Arrange in single layer on baking sheet, working in 2 batches, and broil 6“ from heat until browned, 4 to 5 minutes. Flip and broil until opposite sides are browned and flesh is very tender, 2 to 3 minutes.
Fill large pot with about ½” water and place steamer basket inside. Bring water to a boil over high heat and add Swiss chard, working in 2 batches. Reduce heat to medium-low, cover, and simmer until tender, 4 to 6 minutes. Transfer to colander and drain thoroughly. When cool enough to handle, squeeze chard tightly with hands to extract as much water as possible and blot dry with clean kitchen towel.
Heat oven to 400°F. Coat large skillet with cooking spray and heat over medium-high heat. Add mushrooms and cook, stirring occasionally, until lightly browned and liquid has evaporated, 6 minutes. Reduce heat to medium-low, add garlic and red-pepper flakes, and cook 1 minute. Add Swiss chard and cook until heated through and any liquid has evaporated, 3 minutes. Season to taste with salt and black pepper.
Combine ricotta, egg, and egg whites in large bowl and season with black pepper.
Spread 1 cup of the marinara over bottom of 13“ x 9” baking dish. Add 4 noodles and cover with half the ricotta, then half the eggplant, then half the Swiss chard–mushroom mixture, then ½ cup of the mozzarella, then 1 cup of the marinara. Repeat layers, starting with noodles and ending with 1 cup marinara. Top with a third layer of noodles, remaining 1 cup marinara, and ½ cup mozzarella. Top with Parmesan.
Cover with foil and bake 35 minutes. Remove foil and continue baking until sauce is bubbly and cheese is lightly browned, 15 minutes longer. Rest 30 minutes and serve.