Preheat the oven. Set to 375°F (190°C). Grease a 9x13-inch baking dish with oil or butter.

Sauté the veggies. Heat oil in a large skillet over medium-high. Add peppers and onions with a pinch of salt. Cook 7–9 minutes, stirring occasionally, until softened and lightly browned. Add mushrooms and cook 4–5 minutes more until they release moisture and start to caramelize. Stir in garlic for 30 seconds.

Cook the beef. Push veggies to the side of the skillet. Add a little more oil if needed. Add sliced beef in batches so it sears, not steams. Season with salt and pepper. Cook just until browned; it will finish in the oven. If using deli roast beef, warm it briefly.

Deglaze and season. Splash in Worcestershire sauce and scrape up browned bits. Toss beef and veggies together. Remove from heat.

Make the creamy base. In a bowl, whisk eggs, heavy cream, salt, and pepper. In a separate bowl, mash the softened cream cheese until smooth, then whisk it into the egg mixture until mostly blended. Small specks are fine.

Assemble the casserole. Spread the beef and veggie mixture evenly in the baking dish. Pour the egg-cream mixture over the top, gently shaking the dish to distribute. Layer provolone slices or sprinkle shredded cheese evenly.

Bake. Place on the center rack and bake 20–25 minutes, until the edges are set and the center jiggles slightly. If you like a deeper golden top, broil for 1–2 minutes at the end, watching closely.

Rest and serve. Let it rest 10 minutes so it sets. Slice into squares and garnish with parsley or a dash of hot sauce if you like.