In a large cast iron skillet, place the ground beef and cook, over medium-high heat, until the crust is seared to a dark brown. Stir the beef and continue cooking it until about 50% of the beef is seared. Place a strainer over a large heat-resistant bowl and strain the beef. Set the beef aside.
Add two tablespoons of the beef fat and butter to the skillet. Add the chopped onions, green pepper and mushrooms. Cook, over medium heat, for 1-2 minutes then stir. Cook another 2 minutes until vegetables are just tender.
Add the beef to the vegetables and stir.
In a small mixing bowl, whisk together the cornstarch and beef broth.
Add the beef broth mixture, cider vinegar, ketchup, Worcestershire Sauce, salt, pepper and red pepper flakes to the beef/vegetable mixture.
Bring to a boil, over medium-high heat. Reduce heat to low and simmer until the mixture has thickened, 5-10 minutes.
Heat oven to broil and while the mixture thickens; butter and toast the insides of the buns. Place the buns, buttered side up, on a foil-lined cooking sheet and broil the insides of buns until golden brown. Transfer the buns to a cooling rack.
When the beef mixture has thickened, remove from the heat add the processed cheese. Mix until the cheese has incorporated. Season to taste, if needed.
Remove the top buns from the cooking sheet.
Place a large scoop of the meat mixture on top of each bottom bun and cover with 1-2 slices of Provolone cheese. Broil the sloppy Joes just until the cheese melts and transfer the tray to a cooling rack.
Replace the top buns and serve.