Bring chicken broth to a boil in a large pot then add rice. Cover then turn heat down to medium-low. When rice has 5 minutes left to cook add broccoli and green onions on top to steam.
Meanwhile, heat extra virgin olive oil in a large skillet over medium-high heat. Add chicken, season with salt, pepper, and garlic powder, and then saute until no longer pink. Remove to a plate and set aside then very carefully wipe out skillet with a dry paper towel.
Turn heat down to medium then whisk together flour and 1/2 cup milk in a small bowl. Pour mixture into skillet then add remaining 1-1/2 cups milk. Cook mixture, stirring constantly, until thickened, 7-10 minutes. Remove skillet from heat then add 3/4 of the shredded cheese, sour cream, salt, pepper, and garlic powder, and stir until smooth.
Preheat broiler. Combine cooked rice and broccoli mixture with cheese sauce then scoop mixture into a nonstick sprayed 9x13 baking dish and top with remaining cheese. Broil until cheese is melted then let casserole rest for 5 minutes before serving.
Brown rice requires precise liquid absorption rather than strict timing, so look for tiny steam holes tunneling through the surface to signal that the grain has reached tenderness. Achieving a rich sauce without heavy cream relies on whisking cold milk into flour before applying heat, preventing the starches from clumping into raw pockets. Skillet-seared chicken breasts demand visual monitoring during the sauté phase; slicing a test piece open should reveal opaque white meat without any pink translucency, guaranteeing juiciness without overcooking. Broiling the assembled dish at the final stage requires constant observation because reduced-fat cheeses transition from melted to burned within thirty seconds under direct infrared heat. Adjusting this recipe for smaller households works reliably by halving the grains and liquids while maintaining identical pan dimensions for even moisture evaporation. Letting the finished bake rest for five minutes allows the dairy proteins to set properly, preventing the portions from collapsing into liquid pools upon plating.