Preheat the oven to 350°F and spray a 9x13-inch baking dish with cooking spray; set aside.
In a small saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 minute to form a roux.
Slowly whisk in the chicken broth until the sauce is smooth. Continue to cook until slightly thickened, then remove from heat.
Stir the sour cream and salsa verde into the sauce until combined; set aside.
In a large bowl, combine the chopped cooked chicken and shredded cheddar cheese.
Pour the sour-cream mixture into the chicken and cheese; stir to coat evenly.
Cut each biscuit into 4 to 6 pieces and add them to the bowl. Toss gently so the biscuit pieces are coated with the sauce and mixed with the chicken and cheese.
Spread the mixture evenly in the prepared baking dish, separating any clumps of dough so pieces bake through.
Bake for 25–30 minutes, until the casserole is bubbly and the biscuit pieces are golden brown.
Remove from oven and serve warm; top with extra salsa verde if desired.
