Preheat oven to 400°F (200°C). This higher oven temperature will quickly roast the chicken and give the casserole a nicely browned top during the final bake.

Bring a large pot of salted water to a rolling boil. Cook the pasta 2 minutes shy of package directions (typically 8–9 minutes for penne). Undercooking the pasta keeps it from going mushy in the oven — it will finish cooking with the sauce while baking. Reserve 1 cup of the starchy pasta water, then drain and set the pasta aside.

While the pasta cooks, pat the chicken dry and season on both sides with 1 tsp kosher salt, 1/2 tsp black pepper and 1/2 tsp smoked paprika. Heat 1 tbsp olive oil in a heavy skillet over medium-high heat until shimmering. Sear the chicken 3–4 minutes per side until golden-brown — searing creates Maillard browning which builds flavor. The chicken should be cooked through (internal temperature 165°F/74°C). Transfer to a cutting board to rest 5 minutes, then dice into 1/2-inch cubes. Resting keeps the juices locked in.

Toss the broccoli florets with 1 tsp olive oil, a pinch of salt and roast on a sheet pan in the hot oven for 8–10 minutes, until edges are slightly charred and tender-crisp. Roasting concentrates broccoli’s flavor and prevents it from turning watery in the casserole.

In the skillet used for the chicken, reduce heat to medium, add the remaining 1 tbsp olive oil and 2 tbsp butter. Sauté the diced onion for 3–4 minutes until translucent — cooking the onion softens and sweetens it, building a flavor base. Add the minced garlic and cook 30–45 seconds until fragrant (don’t let it burn).

Sprinkle the flour over the aromatics and stir for 1 minute to cook off the raw flour taste; this creates a roux which will thicken the sauce and help it cling to the pasta and vegetables.

Slowly whisk in the milk, starting with a few tablespoons to loosen the roux, then add the rest in a steady stream. Bring to a gentle simmer while whisking — simmering activates the thickening power of the roux. Add the smoked paprika and a small pinch of nutmeg to round the dairy flavors.

Once the sauce is slightly thickened, turn the heat to low and add the Velveeta in 1-inch cubes, stirring until melted and glossy. Stir in the shredded cheddar until smooth. Using Velveeta gives an ultra-smooth, stable melt and a velvety texture that binds the casserole, while the cheddar contributes punchy classic cheesiness.

Temper the eggs: whisk the 2 large eggs in a small bowl and add 2–3 tablespoons of the warm cheese sauce to the eggs while whisking (tempering prevents the eggs from scrambling when introduced to hot sauce). Then stir the tempered eggs back into the sauce — they add stability and a silky, custardy set to the bake.

Add the rested diced chicken, roasted broccoli, and spinach to the sauce and stir just until the spinach wilts from the heat. If the sauce seems too thick, loosen with reserved pasta water, 1–2 tablespoons at a time — the starchy water helps the sauce cling to the pasta.

Taste and adjust seasoning with remaining salt, pepper and 1 tsp lemon juice if using — acidity brightens the rich cheese sauce and balances the dish.

Combine the cooked pasta and cheese-chicken-veg mixture in a large mixing bowl or directly in a 9x13-inch (or equivalent) casserole dish. Stir gently so pasta is evenly coated.

Make the topping: melt the remaining 2 tbsp butter in a small pan, stir into the panko and grated Parmesan. Scatter the topping evenly over the pasta bake — the buttered panko crisps up in the oven, adding a textural contrast to the creamy interior.

Bake at 400°F (200°C) uncovered for 12–15 minutes, until bubbling and the topping is golden. For an extra-browned finish, place under the broiler 1–2 minutes (watch carefully). Baking sets the custardy egg-cheese matrix and melds flavors.

Let the casserole rest 5–7 minutes before serving so the sauce firms slightly and portions hold their shape. Garnish with a light dusting of extra Parmesan and a grind of black pepper.