In a large skillet, heat the olive oil over medium heat. Add the chopped onion and garlic and sauté them until they are soft and aromatic, about 3-4 minutes.
Increase the heat slightly and add the ground beef. Cook, stirring frequently, until it has browned completely, approximately 5-7 minutes. Drain excess fat if necessary.
Season the beef with ground cumin, smoked paprika, salt, and black pepper. Stir well and let it cook for another 2 minutes to enhance the flavors.
While the beef is cooking, warm the pita pockets in a toaster oven or microwave for a few seconds until soft.
To assemble the pita pockets, spoon the beef mixture into each pita, followed by the shredded lettuce, diced tomato, cucumber, and a sprinkle of feta cheese. Finish with a generous dollop of tzatziki sauce.
Close the pita pockets and serve them immediately for the best taste experience.