In a large skillet over medium heat, cook the ground beef until browned, about 5-7 minutes. Drain excess fat.
Add the chopped mushrooms, onion, and minced garlic to the skillet. Cook until the mushrooms are tender and the onion is translucent, about 5 minutes.
Stir in the oregano, paprika, salt, and pepper. Cook for an additional 2 minutes.
Remove from heat and stir in the shredded mozzarella cheese until melted and well combined.
While the beef mixture cools slightly, prepare the tzatziki sauce. In a bowl, combine the Greek yogurt, grated cucumber, lemon juice, dill, and garlic powder. Mix well and season with salt to taste.
To assemble, gently open each pita pocket and fill with the cheesy beef and mushroom mixture.
Drizzle with tzatziki sauce before serving.