Poach the Chicken: In a large pan on the stove, pour in the chicken broth, minced garlic, and onion wedges. Place the chicken breasts on top of the mixture.
Simmer: Bring to a boil. Once boiling, reduce the heat to medium-low, cover, and cook for 20 minutes.
Shred: Remove the chicken from the pan and shred it into bite-sized pieces. Set the shredded chicken aside, but reserve the cooking broth left in the pan.
Prep the Stuffing: In a large bowl, pour the melted butter over the herb stuffing mix. Toss until the stuffing is evenly coated.
Mix the Sauce: In a separate bowl, use a hand mixer to blend the sour cream, reserved chicken broth, and cream of chicken soup until smooth.
Layer: Lightly grease a 9x13 casserole dish. Press most of the stuffing mix into the bottom of the dish. Layer the shredded chicken on top, followed by the thawed broccoli.
Pour the Sauce: Pour the creamy soup mixture evenly over the chicken and broccoli. Sprinkle the remaining buttered stuffing mix over the very top.
Bake: Bake in a preheated oven at 350°F (175°C) for 30 minutes, or until the top is golden brown and bubbly. Serve warm!