Bring a large pot of salted water to a boil, then add 8 ounces of pasta shells and cook according to package directions.
Add 12 ounces of broccoli florets to the boiling pasta for the last 3 minutes of cooking, then drain both pasta and broccoli in a colander and set aside.
Return the empty pot to medium heat and melt 2 tablespoons unsalted butter.
Whisk in 1/4 cup all-purpose flour and cook, stirring, for 1 minute to form a roux.
Slowly whisk in 2 cups milk until smooth, then bring the sauce to a gentle boil over medium-high heat, stirring until it begins to thicken.
Reduce heat to low and stir in 2 cups shredded sharp cheddar cheese, 1/2 teaspoon onion powder, and 1/2 teaspoon garlic powder until the cheese melts and the sauce is smooth.
Add the drained pasta, broccoli, and 1 pound cooked chicken pieces to the sauce and toss gently to coat evenly.
Season with salt and pepper to taste, then serve immediately.
