Heat the oven to 375°F (190°C) and lightly oil a baking dish. Butterfly the flank steak if needed, place it between sheets of plastic wrap, and pound it to an even ¼- to ½-inch thickness.

Season both sides of the steak with the salt, pepper, Italian seasoning, and optional paprika. Heat the olive oil in a skillet over medium heat, then cook the onion and mushrooms for 5–7 minutes until softened and most of the moisture has evaporated.

Add the garlic and spinach to the skillet and cook for 1–2 minutes, just until the spinach wilts. Cool the vegetable mixture briefly so it does not melt the cheese before the steak is rolled.

Spread the filling over the steak, leaving about a 1-inch border around the edges. Sprinkle the mozzarella or provolone and Parmesan evenly over the vegetables.

Roll the steak tightly from one long side to the other, keeping the grain running lengthwise if possible. Secure the roll with kitchen twine at 1–2-inch intervals or use wooden toothpicks.

For better browning, sear the rolled steak in a hot oven-safe skillet for 2–3 minutes per side. Add the beef broth or red wine if using, then bake uncovered for approximately 20–30 minutes, depending on thickness and desired doneness.

Check the center with an instant-read thermometer and cook the beef to at least 145°F (63°C) for the USDA-recommended minimum. Transfer the steak to a cutting board and let it rest for 10 minutes before removing the twine or toothpicks.

Slice the rested steak into ½- to 1-inch rounds, cutting across the grain for more tender bites. Arrange the slices on a platter and spoon the pan juices over the top before serving.