Preheat a small cast-iron skillet over medium-high heat. Preheat the oven to broil.
Add 2 tablespoons butter to the skillet, then add the onions and season with a pinch of salt and pepper. Cook until softened and beginning to caramelize, 5 to 7 minutes. Remove to a bowl and toss with the balsamic glaze. Set aside.
Sprinkle the filet with the steak seasoning. Melt the remaining 2 tablespoons butter in the skillet. Add the steak and green onions. Cook the steak to your desired doneness, 3 to 4 minutes per side for medium-rare, and blister the green onions, which will take the same time as the steak. Remove both to a cutting board, allowing the steak to rest.
Meanwhile, add the ciabatta rolls to a small sheet pan. Add 1 tablespoon pesto to each half and spread. Sprinkle over the red pepper flakes and grated Parmesan. Top with the mozzarella and red onion. Season each with a pinch of salt and pepper. Place under the broiler to toast and melt the cheese, 2 to 3 minutes. Remove and set aside.
Slice the steak into thin slices and top the toasted rolls. Drizzle with the steak sauce. Chop the green onions and garnish the tops of the pizzas to serve.