Preheat the oven to 350°F (175°C) and spray a 9×13-inch baking dish with cooking spray.
In a medium saucepan, bring 2 cups water to a boil. Stir in 2 cups instant rice, cover, remove from heat, and let stand 5 minutes until water is absorbed.
Meanwhile, in a small saucepan over medium heat melt 3 tablespoons butter. Whisk in 3 tablespoons flour and cook about 1 minute.
Slowly whisk in 2 cups chicken broth until smooth. Cook, stirring, until the sauce thickens, then remove from heat.
Stir the sour cream and 1/2 cup of the salsa verde into the sauce until combined.
In a mixing bowl, combine the cooked shredded chicken, the prepared rice, 1 cup shredded Monterey Jack cheese, and the sour cream sauce; stir until evenly mixed.
Spread the mixture into the prepared baking dish, top with the remaining 1 cup shredded Monterey Jack and the remaining salsa verde.
Bake for 20–25 minutes until heated through and the cheese is melted. If desired, broil a few minutes to brown the cheese—watch closely to avoid burning.