Season the steak evenly with 1 teaspoon of kosher salt and black pepper.
Heat a medium oven-safe skillet over medium-high heat. Add mushrooms to the dry skillet, sprinkle with the remaining ½ teaspoon salt, and cook for about 5 minutes until golden brown.
Increase heat to high, add 2 tablespoons olive oil, then add the onions and bell peppers. Cook for 3–5 minutes until softened. Transfer vegetables to a plate.
Add the remaining 1 tablespoon olive oil to the skillet over high heat. Cook the steak in small batches for 1–2 minutes per side until browned. Transfer to a plate.
Reduce heat to medium, add garlic, and cook for about 1 minute until fragrant.
Return the steak, vegetables, and any juices to the skillet. Stir to combine evenly.
Sprinkle mozzarella and Parmesan evenly over the top.
Turn the broiler on high and place the skillet under the broiler until the cheese is melted and bubbling.
Garnish with green onions if using. Serve warm, either straight from the skillet or inside toasted hoagie rolls.