Arrange onions in an even layer in the bottom of the slow-cooker. Pour beef broth over onions.
Season chicken breasts all over with onion soup mix packet. Add in a single layer to the slow cooker. Sprinkle with the thyme leaves.
Set slow cooker to low and cook for 1 ½ to 2 hours.
When chicken is cooked, preheat oven to broil on high with a rack in the upper third. Transfer chicken, onions, and broth to a baking dish and sprinkle chicken with Swiss. Broil until cheese is bubbly and golden, about 6 minutes. Alternatively, top chicken with cheese in the slow cooker and increase the heat to high. Cover and continue to cook until the Swiss is melted.
Sprinkle with parsley and serve.
