Add the lentils, onion, bell pepper, garlic, broth, tomato sauce, ketchup, brown sugar, Worcestershire, mustard, paprika and salt to a 6-quart slow cooker.
Stir until the ketchup and mustard disappear into the broth.
Cover and cook on low until the lentils are tender but mostly intact, 6 to 6 1/2 hours.
Uncover, stir in the vinegar and switch the cooker to high.
Cook until the mixture mounds on a spoon instead of running, 10 to 15 minutes.
Toast the buns and spoon the lentil filling onto the bottom halves just before serving.