In a large skillet, brown the ground beef, seasoning with salt and pepper as it cooks.
Introduce the diced onions along with both the green and red bell peppers into the skillet. Continue to cook until the vegetables soften and the meat is fully browned.
Eliminate excess grease from the skillet.
Incorporate the Worcestershire sauce and minced garlic, stirring thoroughly.
Mix in the cubed cream cheese, stirring constantly until it melts and combines smoothly with the meat mixture.
Blend in the sour cream followed by the shredded mozzarella cheese, stirring until the cheeses are fully melted and integrated.
Preheat the oven to 350°F.
Lightly coat a baking dish with nonstick spray.
Spoon a portion of the meat mixture into the dish, then layer with provolone or Swiss cheese slices.
Repeat the layering process, finishing with the remaining cheese atop the final meat layer.
Place the dish in the oven and bake for 20 minutes.
Serve warm with hard rolls, crackers, or sturdy chips for dipping.