1. Preheat the oven to 350°F (175°C). Lightly grease a 9×13-inch baking dish and set aside.

2. Bring a large pot of well-salted water to a boil. Add the jumbo pasta shells and cook until just shy of al dente according to package directions. In the last 2 minutes of cooking, add the broccoli florets to the pot. Drain well and spread the shells on a tray so they cool slightly and do not stick together. Set the broccoli aside.

3. In a large mixing bowl, combine the shredded chicken, 1 cup of mozzarella, 2 ounces of the cream cheese, Italian seasoning, onion powder, garlic powder, 1/2 teaspoon of the kosher salt, and 1/4 teaspoon black pepper. Stir until the mixture is creamy and evenly combined, then set aside.

4. In a medium saucepan, melt the butter over medium heat. Sprinkle in the flour and whisk constantly for 1 to 2 minutes to form a smooth roux and cook off the raw flour taste.

5. Add the minced garlic to the roux and cook for about 30 seconds, just until fragrant. Slowly pour in the chicken broth while whisking to avoid lumps, then add the heavy cream. Continue cooking, whisking often, until the sauce thickens slightly and gently bubbles.

6. Reduce the heat to low and whisk in the remaining 2 ounces cream cheese until melted and smooth. Stir in 1 1/2 cups mozzarella and 1/2 cup Parmesan until the cheeses are fully melted and the sauce is creamy. Season with the remaining 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper, tasting and adjusting as needed.

7. Spoon about 1 cup of the cream sauce into the prepared baking dish and spread it into an even layer. Scatter the blanched broccoli over the sauce.

8. Fill each cooked shell with a generous spoonful of the chicken mixture and arrange the stuffed shells in snug rows over the broccoli, open side up.

9. Pour the remaining cream sauce evenly over the shells, making sure each one gets some sauce. Sprinkle the top with the remaining 1 1/2 cups mozzarella and 1/2 cup Parmesan. If you like, add an extra pinch of Italian seasoning over the cheese.

10. Bake for 25 to 30 minutes, or until the casserole is bubbling around the edges and the cheese is melted and lightly golden. For a deeper brown top, broil for 1 to 2 minutes at the end, watching closely.

11. Remove the dish from the oven and let the cheesy chicken shells rest for about 10 minutes so the sauce can thicken slightly. Garnish with chopped fresh parsley, then serve warm.