Add chicken broth to a large pot then bring to a boil and add rice. Cover then turn heat down to low and cook according to package directions. When rice has 5 minutes left to cook, add broccoli and green onions (no need to stir) then replace lid and continue cooking until rice and vegetables are tender.
Meanwhile, heat extra virgin olive oil in a large skillet over medium-high heat. Add chicken, season with salt, pepper, and garlic powder, then saute until cooked through. Remove to a plate then set aside. Turn heat down to medium then very carefully wipe skillet out with a dry paper towel.
In a small dish, whisk together flour and 1/2 cup milk then pour mixture into skillet. Add remaining 1-1/2 cups milk then cook mixture while 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 then stir until smooth.
Preheat broiler. Combine cooked rice and broccoli mixture with the cheese sauce and cooked chicken 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.