Lightly grease the inside of the slow cooker insert with non-stick cooking spray.
Place the chopped green cabbage wedges evenly across the bottom of the slow cooker.
Scatter the sliced yellow onion, minced garlic, and sliced smoked sausage directly over the top of the cabbage.
Pour the chicken broth evenly over the ingredients. Sprinkle with the salt, black pepper, and dried thyme.
Place the small cubes of butter randomly across the top of the meat and vegetables.
Cover the slow cooker and cook on LOW for 5-6 hours, or on HIGH for 3-4 hours, until the cabbage is incredibly tender.
Using a stainless steel spatula, toss the ingredients gently before serving to mix the savory pan juices and melted butter over the cabbage and sausage.