Spray a 8x8" baking dish (or equivalent-sized baking dish) very well with nonstick spray then set aside. Bring chicken broth and salsa verde to a boil in a medium-sized saucepan then add rice, place a lid on top, turn heat down to low, then cook according to package directions. Remove pan from heat then fluff rice, sprinkle green onions on top, and then place the lid back on the pan for 5 minutes.
Meanwhile, heat extra virgin olive oil in a large, deep skillet over medium-high heat. Add chicken and green chilies, season with salt and pepper, and then saute until chicken is just cooked through. Add garlic then saute for 1 more minute. Remove chicken and chili mixture to a plate then very carefully wipe out the inside of the skillet with a couple of paper towels. Let the skillet cool off the heat while you whisk the flour into a dish with 1/2 cup milk.
Return the skillet to the burner over medium-high heat, add the flour and milk mixture, and then add the remaining 1-1/2 cups milk. Season with salt and pepper then let the mixture come to a simmer, stirring constantly. Turn heat down to medium then simmer until thickened, 8-10 minutes, stirring constantly. Remove the skillet from the heat then stir in 6oz shredded cheese and sour cream. Taste then add salt and pepper if necessary.
Add the chicken mixture and rice to the cheese sauce then stir to combine and scoop into the prepared baking dish. Top with remaining cheese then broil until golden brown.