Heat oil in a large skillet over medium-high heat. Add carrots and onion; cook, stirring occasionally, until starting to brown, about 8 minutes. Stir in garlic and chili powder; cook, stirring, until fragrant, about 15 seconds. Remove from heat; stir in vinegar and scrape up any browned bits.
Coat a 6-quart slow cooker with cooking spray. Drain and rinse the soaked beans; transfer to the slow cooker. Stir in bell pepper, tomato sauce, water, soy sauce (or tamari) and tomato paste to combine.
Spread the carrot-onion mixture over the bean mixture, but do not stir the two together. (The layer on top helps keep the beans submerged during cooking so they do not dry out.) Cover and cook on High for 5 hours or Low for 9 hours.
Stir in cabbage, zucchini, corn, honey mustard, brown sugar and salt; cook on High for 30 minutes.
Serve the mixture on buns and enjoy!