Bring a large pot of water to a rolling boil and salt it generously (about 1 tbsp kosher salt). The salted boiling water seasons the pasta from the inside out. Add the penne and cook to 2 minutes shy of package al dente (usually 8–9 minutes) because it will finish cooking in the oven. Reserve 1/2 cup of the pasta cooking water, then drain the pasta and set aside.

While the pasta cooks, butterfly the chicken breast: lay it flat and slice horizontally almost to the edge, then open it like a book. Slice the chicken into 1/2-inch strips. This speeds cooking and makes shredding/tender slices easier, giving consistent pieces in the bake.

Heat a large skillet over medium-high heat and add the olive oil. When the oil shimmers (this indicates it's hot enough to sear), add the ground beef. Brown the beef without overcrowding so it can caramelize — stir occasionally and break into bite-sized pieces. Browning creates Maillard flavors (deep savory notes) that make the sauce rich. Cook about 6–8 minutes until nicely browned; transfer beef to a plate and leave any rendered fat in the pan.

Add the butter to the same hot skillet with remaining beef fat, then add the sliced chicken. Season lightly with a pinch of salt and pepper. Sear the chicken for 2–3 minutes per side until golden but slightly underdone; it will finish cooking in the tomato sauce and oven. Removing the chicken when slightly underdone keeps it tender rather than rubbery.

Push the chicken to one side of the pan (or remove and return later) and add the diced onion to the cleared space. Sauté for 3–4 minutes until translucent — sweating the onion releases its sweet, savory base flavors. Add the minced garlic and cook 30 seconds until fragrant; garlic burns quickly, so brief cooking preserves its flavor.

Stir in the crushed tomatoes, chicken broth, dried oregano, red pepper flakes, remaining 1/2 tsp kosher salt, and 1/4 tsp black pepper. Return the browned ground beef to the pan and nestle the seared chicken back in. Bring to a gentle simmer and let cook 6–8 minutes so flavors meld and chicken finishes cooking. Simmering concentrates flavors and allows the meat to absorb the tomato herb notes.

Remove the chicken to a cutting board and shred or thinly slice it. Stir the milk into the sauce — milk softens the acidity of tomatoes and gives a silkier mouthfeel. If the sauce seems too thick, add up to the reserved 1/2 cup pasta water to loosen; the starchy water helps the sauce cling to the pasta.

Preheat the oven to 400°F (200°C). In a large mixing bowl, combine the par-cooked penne with the tomato-meat sauce, shredded/sliced chicken, and half of the mozzarella and half of the Parmesan. Toss to coat evenly so every piece of pasta is flavored; this prevents dry pockets in the bake.

Transfer the pasta mixture to a 9x13-inch (or similar) baking dish, smooth the top, and sprinkle the remaining mozzarella and Parmesan evenly over the surface. The cheese top forms a golden, flavorful crust that contrasts the saucy pasta beneath.

Bake uncovered for 18–22 minutes until the cheese melts and begins to brown. For a blistered, restaurant finish, switch the oven to broil for 1–2 minutes watching closely — broiling quickly browns and crisps the cheese (don’t walk away; it can burn in seconds).

Let the casserole rest 5 minutes (this helps the sauce set so it doesn’t run). Sprinkle chopped parsley before serving for brightness. Serve hot, with an optional extra grind of black pepper.