Heat a large, 12-inch nonstick skillet over medium heat and add the meat, onions, bell pepper, mushrooms, salt and pepper. Cook, stirring often to break the meat into small pieces, until the meat is evenly and completely cooked and the vegetables are tender, 5-10 minutes. Drain excess grease.
Sprinkle the flour over the meat/veggies and stir to combine. Cook the mixture over medium heat for 1-2 minutes, stirring often (this helps cook out the flour taste).
In a bowl or liquid measuring cup, whisk together the beef broth, Worcestershire sauce, and BBQ sauce.
Over medium heat, gradually add the sauce mixture to the skillet, stirring with a wooden or silicone spoon or spatula. Bring the mixture to a simmer and cook for 4-5 minutes until bubbly and thickened. Add additional salt and pepper to taste, if needed.
Serve the sloppy joe mixture on buns topped with a slice of provolone cheese (or other cheese of choice). For a meltier cheese, place the assembled sloppy joes in a 350 preheated oven for a few minutes before serving.
