PREPARE: Preheat oven to 350˚ Brush a light layer of olive oil an 9×13 baking dish. Rinse the quinoa in a mesh sieve until the water runs clear.
COOK THE QUINOA & BROCCOLI: Pour the quinoa into a 3 qt saucepan with 2 cups of water. Bring the water to a boil, then reduce to a simmer and cover. Continue to simmer for 12 minutes.
Add the broccoli and an additional ½ cup of water to the pot, over the top of the quinoa and continue to steam an additional 5 minutes.
COOK THE CHICKEN: While the quinoa cooks, heat 1 tablespoon olive oil in the skillet over medium heat. Season the chicken with salt and pepper, to taste.
Once the oil is hot, add the chicken to the skillet, cook 5-6 minutes per side or until the center of each reads 160˚F with an instant read thermometer. Allow the chicken to cool. MAKE THE CHEESE SAUCE: Melt butter in the skillet over medium heat, add the onion, thyme and garlic to the pan and cook for 4 minutes or until the onions are soft. Whisk in flour until golden, about 1 minute.
Gradually whisk in milk, and whisk constantly, until slightly thickened, about 3-4 minutes. Remove the pan from the heat and add 1 ¼ cup of shredded cheese. Whisk until the cheese melts and the sauce is smooth.
ASSEMBLE THE CASSEROLE: Chop the chicken breasts into 1-inch pieces, add it to a large bowl along with the quinoa, broccoli and cheese sauce. Fold to evenly coat everything with the cheese sauce. Spread the mixture evenly in the prepared baking dish. Top with remaining ¾ cup cheese and parmesan. Place into oven and bake until cheese has melted, about 8 minutes. Serve right away.