Bring a large pot of salted water to a boil and cook the shell pasta according to package directions; add the diced broccoli during the last 3–4 minutes of cooking. Reserve 1/4 cup pasta cooking water, then drain pasta and broccoli well in a colander.
Meanwhile, melt the butter in a medium saucepan over medium heat. Stir in the flour and cook, stirring constantly, for 1 minute to form a roux.
Whisking continuously, slowly pour in the milk. Add the onion powder, garlic powder, and salt and pepper to taste. Increase heat to medium-high and cook, stirring constantly, until the sauce comes to a boil and thickens.
Remove the sauce from the heat and let it cool 1 minute, then stir in the shredded cheddar and parmesan until fully melted and smooth.
Add the drained pasta and broccoli plus the shredded chicken to the cheese sauce and toss gently to coat evenly. If the sauce becomes too thick as it sits, thin with a few tablespoons of the reserved pasta water.