In a slow cooker, combine the beef chunks, sliced onion, and minced garlic.
In a small bowl, mix together the olive oil, oregano, cumin, smoked paprika, salt, and black pepper. Pour this mixture over the beef and onion.
Add the beef broth and red wine vinegar to the slow cooker. Stir gently to combine.
Cover and cook on low for 8 hours or on high for 4 hours, until the beef is tender and easily shredded with a fork.
Once cooked, shred the beef directly in the slow cooker using two forks, mixing it with the juices.
Serve the shredded beef in warm pita breads, topped with tzatziki sauce, diced tomatoes, shredded lettuce, and sliced red onion.