Preheat oven to 350 degrees. Spray an 9x13 baking dish with nonstick spray then set aside.
In a medium saucepan, over high heat, bring 2 cups low sodium chicken broth to a boil. Stir in the rinsed and drained quinoa. Cover, reduce heat to low, and simmer for about 20 minutes or until almost all the liquid has been absorbed and quinoa is tender. Remove from heat then let sit, covered, for 10 minutes and set aside.
In a large microwave-safe bowl, add the broccoli florets and 1/4 cup of water. Heat in the microwave for 2 minutes until the broccoli florets are bright green and slightly tender. Set aside.
Meanwhile, in a large skillet or dutch oven over medium-high heat, add 1 tablespoon of olive oil and melt one tablespoon of butter. Add chopped onion and chopped celery and saute for about 5 minutes until the onions are soft. Add minced garlic and saute for about 1 minute.
Sprinkle in flour (gluten-free flour, if desired) and stir to cook for 1 minute. Reduce heat to medium. Slowly pour in 1 cup low sodium chicken broth, followed by one cup of milk, stirring to avoid lumps. Add poultry seasoning, and salt and pepper to taste. Turn heat back down to low then simmer until slightly thickened, 5 minutes.
Remove skillet from heat then pour sauce into the large bowl of broccoli. Add cooked quinoa, cooked chicken, and half the cheese and then stir to combine.
Pour mixture into prepared 9x13 baking dish then sprinkle remaining cheese on top. Sprinkle bread crumbs evenly on top. Place the baking dish on top of a baking sheet in case of bubble overs then bake for 20-25 minutes or until top is golden brown and bubbly. Let cool for 10 minutes before serving.