Preheat your oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish with nonstick cooking spray.
In a large pot or Dutch oven, bring the chicken broth to a gentle simmer over medium heat. Add the sliced yellow squash and cook for 8 minutes until it begins to soften.
Drain the squash well to remove excess moisture and transfer it to a large mixing bowl. Add the shredded carrots, chopped onion, and shredded cooked chicken. If you are cooking raw chicken for this step, ensure it reaches a safe internal temperature of 165°F (74°C) before shredding.
In a separate smaller bowl, stir together the sour cream, cream of mushroom soup, kosher salt, and black pepper until smooth.
Pour the soup mixture over the chicken and vegetables. Stir gently until all the ingredients are evenly coated.
Spoon the mixture into the prepared baking dish and spread it into an even layer.
In a small bowl, toss the dry stuffing mix with the melted butter until the crumbs are moistened. Sprinkle this mixture evenly over the top of the casserole.
Bake uncovered for 25 minutes, or until the edges are bubbly and the stuffing is golden brown. Let the casserole rest for 5 minutes before serving.