Cook the vegetables. Heat a large skillet over medium-high heat with a drizzle of oil. Add the diced onions and bell peppers and cook for 3–4 minutes until softened. Add the mushrooms and garlic, cooking another 2–3 minutes until the mushrooms release their moisture and start to brown.

Brown the beef. Push the veggies to the side and add the ground beef to the pan. Break it apart and cook until no longer pink, about 5–6 minutes. Drain off excess fat if needed, then stir everything together.

Season the filling. Add the Worcestershire sauce, soy sauce, steak seasoning, smoked paprika, salt, and pepper. Stir well and let the mixture simmer on low while you make the cheese sauce.

Make the cheese sauce. In a small saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 minute. Slowly pour in the milk, whisking constantly until the sauce thickens, about 2–3 minutes. Remove from heat and stir in the shredded provolone until smooth.

Assemble. Spoon the meat mixture generously onto toasted buns and drizzle the warm cheese sauce right over the top. Serve immediately!