Place onions in a 3 1/2- to 5-quart slow cooker. Trim separable fat from beef. If necessary, cut beef to fit cooker. Place beef on top of onions. Sprinkle with garlic, thyme, and pepper. Pour beef broth, water, and Worcestershire sauce over all.

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.

Add red pepper to the slow cooker to soften slightly. Transfer meat to a cutting board; thinly slice across the grain, removing any visible fat as you slice. Using a slotted spoon, remove onions and red pepper from cooker. Divide sliced brisket and onion and pepper slices among bread bottoms. Add bread tops. Skim fat from cooking juices in cooker; pass juices for dipping sandwiches.