Sauté the vegetables. Melt 2 tablespoons of butter in a skillet over medium heat. Add the onion, celery, and mushrooms and cook for 5–7 minutes, until softened and most of the moisture from the mushrooms has cooked away. Add the garlic and cook for another 30 seconds, until fragrant.
Add the chicken. Arrange the chicken breasts in the bottom of a slow cooker.
Make the creamy filling. Stir together the cream of chicken soup, sour cream, Worcestershire sauce, and seasonings. Spread sautéed vegetables evenly over the chicken. Then add creamy filling on top. Do not add any broth at this point—the chicken will release liquid as it cooks.
Slow cook. Cover and cook on LOW for 3–4 hours, beginning to check the chicken around the 3-hour mark. (my slow cooker runs hot so I did one hour on high and one hour on low and it was nice and tender.) Cook until the thickest part of the chicken reaches 165°F.
Shred the chicken. Remove the cooked chicken and shred or chop it into bite-sized pieces. Return it to the slow cooker and stir it into the creamy sauce along with the canned grean beans
Prepare the stuffing. In a bowl, toss the dry stuffing mix with ½ cup low-sodium chicken broth and 2 tablespoons melted butter. The stuffing should be evenly but lightly moistened, not saturated.
Add the stuffing. Spread the stuffing loosely and evenly over the chicken mixture. Don't stir it into the filling.
Finish cooking. Cover and cook on HIGH for 20–30 minutes, or until the stuffing is tender and the mixed vegetables are heated through.
Serve. Let the casserole sit uncovered for about 5 minutes before serving to allow the creamy filling to thicken slightly.