Spray the inside of a 4-6 quart slow cooker with non-stick cooking spray.
Place the cut chicken breasts evenly in the bottom of the slow cooker.
Add the chopped onion, minced garlic, and fresh broccoli florets on top of the chicken.
In a medium bowl, whisk together the condensed cream of chicken soup, sour cream, chicken broth, dried thyme, salt, and black pepper until smooth.
Pour the soup mixture evenly over the chicken and vegetables in the slow cooker.
Sprinkle the shredded cheddar and mozzarella cheeses evenly over the top.
Cover and cook on LOW for 4 hours, or until the chicken is cooked through and the broccoli is tender.
In a small bowl, combine the plain breadcrumbs with the melted butter until the crumbs are evenly coated.
Carefully sprinkle the breadcrumb mixture over the top of the casserole.
Cover and cook on HIGH for an additional 15-20 minutes to allow the breadcrumb topping to crisp up.
Turn off the slow cooker and let the casserole rest for 5 minutes before serving.