Bring a large pot of salted water to a boil, then add 8 ounces pasta shells and cook according to package directions.
Add 12 ounces broccoli florets to the boiling pasta during the last 3 minutes of cooking; drain 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 1 minute to form a roux.
Slowly whisk in 2 cups milk until smooth, then continue whisking over medium-high heat until the sauce begins to thicken.
Remove the sauce from heat and stir in 2 cups shredded sharp cheddar cheese, 1/2 teaspoon onion powder, and 1/2 teaspoon garlic powder until melted and combined.
Add the cooked pasta, broccoli, and 1 pound bite-sized cooked chicken to the cheese sauce and toss to coat evenly.
Season with salt and pepper to taste and serve immediately.