Heat olive oil in a large frying pan over medium heat. Add onion and carrots and saute until they begin to soften, about 5 minutes. Add mushrooms, cumin, and paprika. Stir everything together and allow mushrooms to soften, 2 to 3 minutes.
Add beans, tomato sauce, vinegar, mustard and syrup, and allow to simmer and thicken for about 15 minutes. Taste and add salt and pepper if you like.
Toast buns (to make Joes a bit less sloppy). Spoon a generous amount of bean mixture onto the bottom half of each bun and sprinkle with a good pinch of shredded cheese. Put hamburger lid on top and serve. Makes 6 servings.