Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish with cooking spray or a little oil.
In a large skillet over medium-high heat, cook the ground beef, breaking it up with a spoon, until no longer pink, about 5-7 minutes. Drain any excess fat. Add the sliced onions, bell peppers, and mushrooms to the skillet. Cook, stirring occasionally, until the vegetables have softened and the onions are translucent, about 8-10 minutes. Stir in the minced garlic, Worcestershire sauce, smoked paprika, salt, and pepper. Cook for 1 more minute until fragrant.
In a small bowl, whisk together the beef broth and cornstarch until smooth. Pour the mixture into the skillet with the beef and vegetables. Bring to a simmer, stirring constantly, until the sauce thickens slightly, about 2-3 minutes. Remove from heat.
Spread the thawed hash browns evenly in the bottom of the prepared baking dish. Pour the entire beef and vegetable mixture over the hash browns, spreading it into an even layer. Sprinkle 1 1/2 cups of the shredded provolone cheese evenly over the top.
In a small bowl, mix the remaining 1/2 cup provolone with the mozzarella and Parmesan. Sprinkle this cheese blend evenly over the casserole. Bake, uncovered, in the preheated oven for 25-30 minutes, or until the cheese is melted, bubbly, and starting to turn golden in spots.
Remove the casserole from the oven and let it rest for 5-10 minutes before serving. This allows the layers to set for cleaner scoops. Garnish with chopped fresh parsley if desired.