Set a large, deep skillet or cast-iron pan over medium-high heat and melt the 2 tablespoons of unsalted butter. Drop in the finely diced sweet yellow onion, green bell pepper, and chopped mushrooms, cooking for 4 to 5 minutes while stirring frequently until the vegetables soften and take on light, golden caramelized edges.

Stir in the minced garlic and cook continuously for exactly 60 seconds until intensely fragrant, taking care not to let the small garlic pieces burn.

Crumble the 1 pound of lean ground beef directly into the hot vegetable skillet. Cook for 5 to 6 minutes, using a firm wooden spoon or spatula to aggressively chop the meat into uniform, fine bits until it is completely browned and no pink remains. Tilt the pan slightly to ladle out or drain away any heavy pooling grease.

In a separate small bowl or measuring cup, pour in the 1 cup of low-sodium beef broth and whisk in the 1 tablespoon of cornstarch until completely dissolved and lump-free.

Pour the beef broth slurry, ketchup, and Worcestershire sauce straight into the bubbling meat skillet, stirring firmly to scrape up any savory browned bits stuck to the bottom of the pan. Season the mixture with salt and black pepper to taste.

Bring the liquid up to a boil, then immediately turn the stove burner down to medium-low to maintain a gentle, open simmer. Cook uncovered for 4 to 5 minutes, stirring occasionally, until the starches hydrate and the liquid reduces into a thick, glossy, and savory gravy that snugly envelopes the beef.

Turn off the heat. If you prefer your cheese completely folded into the filling, scatter the shredded cheese over the hot meat now and stir gently for 30 seconds until entirely liquefied and stringy.

Alternatively, split your hamburger buns and arrange them on a baking sheet. Scoop mounds of the saucy beef filling onto the bottom buns, blanket each sandwich with a slice of provolone cheese, and place under your oven's broiler for 1 to 2 minutes until the cheese is entirely melted, bubbling, and golden. Crown with the top buns and serve piping hot.