Brown the ground beef with the chopped onion and minced garlic in a large skillet over medium-high heat until the meat is no longer pink; drain excess fat.
Place the chopped cabbage into the bottom of a 6-quart slow cooker.
Add the browned beef mixture, uncooked rice, crushed tomatoes, tomato sauce, beef broth, brown sugar, Worcestershire sauce, salt, pepper, and dried thyme to the slow cooker.
Stir the ingredients thoroughly to ensure the rice is submerged in the liquid and the components are evenly distributed.
Cover and cook on low heat for 6 to 8 hours or on high heat for 3 to 4 hours, until the cabbage is tender and the rice is translucent and soft.
Stir the mixture thoroughly before serving to incorporate any liquid at the bottom.