Preheat oven to 375°F (190°C). Grease a 9x13-inch baking dish with the butter. Preheating ensures the oven is at the right temperature for even baking so the top crisps while the interior sets.

Bring a large pot of salted water to a rolling boil. Cook the pasta 2 minutes less than package directions for al dente (it will finish cooking in the oven). Undercooking the pasta prevents it from turning mushy after baking because it will absorb sauce and continue to cook.

While pasta water heats, season the chicken breasts with 1/2 tsp salt and a few grinds of pepper. Heat 1 tablespoon olive oil in a skillet over medium-high heat and sear the chicken 3–4 minutes per side until golden but not fully cooked through. Transfer to a plate and cover loosely with foil to rest. Searing develops flavor through Maillard browning and partially cooking speeds shredding later.

In the same skillet, add the remaining 1 tablespoon olive oil. Add the chopped onion and a pinch of salt; sauté over medium heat until translucent and lightly golden, about 5–7 minutes. Add the garlic and cook 30–45 seconds until fragrant. Cooking the onion slowly builds sweetness and depth; adding garlic later prevents it from burning and turning bitter.

Add the ground beef to the skillet with the onion and garlic. Break it up with a spoon and cook over medium-high heat until well-browned and no longer pink, about 6–8 minutes. Browning the beef (rather than just cooking) creates concentrated savory flavor from caramelized meat bits fond.

Stir in the tomato paste and cook 60 seconds to caramelize it — this blooms tomato flavor. Add the crushed tomatoes, oregano, dried basil, red pepper flakes (if using), and sugar. Season with 1 tsp salt and 1/2 tsp black pepper. Bring to a gentle simmer, reduce heat to medium-low, and cook uncovered 12–15 minutes until the sauce thickens slightly. Simmering melds flavors and concentrates the sauce so it won’t make the bake watered-down.

While the sauce simmers, shred the seared chicken: either use two forks to pull it apart or shred with a stand mixer paddle on low for 30 seconds. Return the shredded chicken to the sauce for the last 5 minutes of simmering to absorb flavor and finish cooking through (carryover heat will complete it). Adding cooked shredded chicken at the end keeps its texture tender and prevents it from drying out.

Reserve 1 cup of the simmered sauce in a small bowl for topping. Taste the remainder and adjust seasoning. Stir the cooked (but still slightly firm) pasta into the large bowl and toss with most of the sauce (reserve a little for dryness), leaving enough sauce to coat but not drown the pasta. Tossing while hot allows the sauce to cling to the pasta ridges, giving you even flavor in every bite.

In a separate bowl whisk together the eggs, milk, and half (about 3/4 cup) of the shredded mozzarella plus half of the grated Parmesan. Whisking the eggs with dairy creates a loose custard that, when baked, binds the pasta and meats into a creamy, sliceable interior instead of a dry casserole.

Combine the sauced pasta, the remaining unblended shredded beef/chicken mixture (ensure meat is evenly distributed) and the egg-cheese mixture in a large bowl; toss gently to combine. The goal is an even distribution of meat, cheese and pasta so each serving has balanced flavor and texture.

Transfer the mixture to the prepared 9x13-inch baking dish. Smooth the top with a spatula. Sprinkle the remaining mozzarella and Parmesan evenly over the top, and spoon the reserved cup of sauce in dots across the surface (this gives pockets of bright tomato). A smooth top allows the cheese to brown evenly; the dotted sauce creates bright contrast and prevents the surface from drying.

Bake uncovered for 20–25 minutes, until the cheese is melted and the edges are bubbling. If you like a more golden, blistered top, turn the oven to broil for 1–2 minutes while watching closely to avoid burning. Baking sets the egg custard and develops a gratin-like crust that contrasts with the creamy interior.

Let the bake rest 8–10 minutes before serving. Resting lets the custard firm up so you can slice cleanly and the flavors marry. Garnish with torn fresh basil leaves and an extra sprinkle of Parmesan before serving.