Preheat your oven to 375°F (190°C). Cook pasta to al dente according to package directions. Blanch broccoli florets in boiling water for 2-3 minutes, then transfer to an ice bath. Drain pasta and broccoli well.

In a large saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 minute. Gradually whisk in the milk until smooth. Cook until the sauce thickens, stirring often. Remove from heat and stir in Dijon mustard, garlic powder, onion powder, salt, pepper, and 1 cup of the cheddar cheese until melted and smooth.

In a large mixing bowl, combine the cooked pasta, blanched broccoli, and shredded chicken. Pour the cheese sauce over the mixture and stir until everything is well-coated.

Transfer the mixture into a 9x13 inch baking dish. Top with the remaining 1 cup of cheddar cheese and the mozzarella cheese. Bake for 20-25 minutes, until the casserole is bubbly and the cheese is golden.

Let the casserole rest for 5-10 minutes before serving. This allows it to set for easier serving.