Heat olive oil in a large skillet or Dutch oven over medium-high heat. Add ground beef and cook, breaking it apart with a spatula, until browned throughout. Drain any excess fat and set the beef aside.

In the same skillet, add the chopped onion, green bell pepper, and sliced mushrooms. Sauté for 7-10 minutes, or until the vegetables have softened and released their moisture.

Stir in the minced garlic and cook for another minute until fragrant.

Return the cooked ground beef to the skillet with the vegetables. Add the Worcestershire sauce, beef broth, ketchup, Dijon mustard, brown sugar, Italian seasoning, salt, and black pepper. Stir everything to combine.

Bring the mixture to a simmer, then reduce the heat to low, cover, and let it cook for 10-15 minutes, stirring occasionally, to allow the flavors to meld and the sauce to thicken slightly.

While the sloppy joe mixture simmers, preheat your oven to 350°F (175°C) or a broiler. Slice the hoagie rolls lengthwise if not already cut. You can lightly toast them in a dry skillet, under the broiler for 1-2 minutes, or in the oven for 3-5 minutes until golden.

Spoon a generous amount of the Philly cheesesteak sloppy joe mixture onto the bottom half of each toasted hoagie roll.

Place a slice of provolone or American cheese over the sloppy joe mixture.

Place the assembled sandwiches on a baking sheet (if using oven/broiler) or place the top bun on the cheese (if melting with residual heat). If using the oven/broiler, return them to the oven or under the broiler for 1-2 minutes, or until the cheese is melted and bubbly.

Serve immediately and enjoy your ultimate Philly Cheesesteak Sloppy Joes!