Preheat the oven to 450°F. Warm 1 tablespoon of the olive oil in a large frying pan over medium-high heat. Add the mushrooms, season with salt and pepper, and cook, stirring often, until soft, about 4 minutes. Transfer to a plate; do not wipe the pan clean. Warm 1 tablespoon of the olive oil in the same pan and add the spinach. Cook, stirring constantly, until the spinach just begins to wilt, about 2 minutes. Transfer to a separate plate and set aside. Wipe the pan clean and reserve. 

Working with 1 chicken breast at a time, place the chicken between 2 pieces of plastic wrap and pound with the flat side of a meat pounder to a thickness of about ¼ inch. Pour the flour onto a large plate and season it well with salt and pepper. Pour the bread crumbs onto another plate and season with salt and pepper. Lightly beat the eggs with 1 tablespoon water in a shallow bowl. 

In the reserved frying pan, heat the remaining 2 tablespoons olive oil over medium­-high heat until very hot but not smoking. Working with 1 at a time, lightly dredge the pounded chicken pieces first in the seasoned flour; then in the egg wash, letting the excess drip off; then in the bread crumbs. When the oil is very hot but not smoking, working in batches as needed to avoid crowding the pan, add the cutlets and fry, turning once, until golden and crusty on both sides, about 4 minutes per side. 

Ladle the marinara sauce in a baking dish. Transfer the chicken cutlets as they are finished to the baking dish and top with the spinach and mushrooms. Arrange the mozzarella slices on top and sprinkle with the Parmesan. Bake until the chicken is opaque throughout and the cheese is melted, about 15 minutes. Serve right away.