Preheat oven to 350°F with rack in top third position. Lightly coat a 9-by-13-inch baking dish with cooking spray.

Combine 2 cups water and 1 cup quinoa in a large saucepan; bring to a boil over medium-high heat. Reduce heat to medium-low; cover and cook for 10 minutes. Add broccoli pieces (do not stir); cover and continue cooking until the water is absorbed, 3 to 5 minutes. Remove from heat and let stand, covered, for 5 minutes. Transfer the mixture to the prepared baking dish.

Meanwhile, heat 1 tablespoon oil in a large skillet over medium-high heat. Season cubed chicken with 1 teaspoon garlic-and-herb seasoning and ¼ teaspoon each salt and pepper. Add to the skillet; cook, undisturbed, until golden on one side, about 6 minutes. Transfer the chicken to the baking dish and stir to combine. Do not wipe the skillet clean.

Add sliced scallions and 2 tablespoons butter to the skillet. Cook over medium heat, stirring occasionally, until the scallions are bright green and the butter is melted, about 30 seconds. Stir in 2 tablespoons flour, 2 teaspoons thyme and the remaining 3 teaspoons garlic-and-herb seasoning and ½ teaspoon salt. Cook, stirring constantly, until fragrant, about 1 minute. Gradually whisk in 2 cups milk; cook, stirring often, until thickened, about 5 minutes. Remove from heat and stir in 1 cup cheese until melted.

Pour the sauce over the quinoa mixture in the baking dish and stir to combine. Sprinkle with the remaining 1 cup cheese.

Cover with foil and bake until the chicken is cooked through (an instant-read thermometer inserted in a piece of chicken registers 165°F), 15 to 20 minutes. Remove from oven and increase oven temperature to broil. Discard the foil and broil until the cheese is melted and lightly browned in spots, about 5 minutes. Garnish with thyme leaves, if desired.