Place onion rings in a 3-1/2- to 5-quart slow cooker. Trim fat from brisket. If necessary, cut brisket to fit in cooker. Place brisket on top of onion. Sprinkle with garlic, thyme, and pepper. Pour beef broth and Worcestershire sauce over all in cooker.
Cover and cook on low-heat setting for 9 to 10 hours for brisket or 8 to 9 hours for bottom round or on high-heat setting for 4-1/2 to 5 hours for brisket or 4 to 4-1/2 hours for bottom round.
Remove meat from cooker, reserving cooking liquid in cooker. Thinly slice meat across the grain, removing visible fat as you slice. Using a slotted spoon, remove onion from cooker. Place 3 to 4 ounces sliced brisket on top of each bread bottom. Top each with about 1/3 cup onion. Add bread tops. Skim fat from cooking liquid in cooker; pass liquid for dipping sandwiches (about 1/2 cup per serving).