In a large skillet over medium-high heat, cook the ground beef, breaking it up into small crumbles, until browned and no longer pink, about 6 to 8 minutes. Drain off excess grease.
Transfer the cooked beef to the slow cooker. Add the diced onion, green bell pepper, and minced garlic.
In a bowl, whisk together tomato sauce, tomato paste, brown sugar, worcestershire sauce, yellow mustard, apple cider vinegar, chili powder, smoked paprika, salt, and black pepper. Pour the sauce into the slow cooker and stir well to combine.
Cover and cook on low for 5 to 6 hours or on high for 2.5 to 3 hours, until the vegetables are tender and the flavors have melded together.
Remove the lid and stir the mixture. If the sauce is too thin, leave the lid off and cook on high for 15 to 20 minutes until it thickens slightly.
Spoon the sloppy joe mixture generously onto the toasted hamburger buns and serve immediately.