Heat a large skillet over medium-high heat.
Add the ground beef to the skillet and cook, breaking it apart with a spatula, until browned and no longer pink, about 5-7 minutes.
Drain excess fat from the skillet.
Add the chopped onion, green bell pepper, and minced garlic to the skillet with the beef. Cook, stirring occasionally, until the vegetables are softened, about 4 minutes.
Reduce heat to medium and stir in ketchup, tomato paste, Worcestershire sauce, apple cider vinegar, brown sugar, yellow mustard, smoked paprika, salt, and black pepper.
Simmer the mixture, stirring occasionally, until thickened and heated through, about 8-10 minutes.
While the sauce simmers, lightly toast the hamburger buns if desired.
Spoon the sloppy joe mixture onto the bottom halves of the buns and cover with the top halves.
Serve immediately.