In a small bowl, combine Greek yogurt, cucumber, garlic, vinegar, lemon juice, salt and pepper, to taste; drizzle with olive oil. Refrigerate for at least 30 minutes, allowing the flavors to meld; set aside.
In a large bowl, combine garlic, lemon juice, red wine vinegar, 2 tablespoons olive oil, yogurt, oregano, salt and pepper, to taste. Add sirloin, coating the beef completely; cover and place in the refrigerator for at least one hour.
Heat remaining 1 tablespoon olive oil in a cast iron skillet over medium high heat. Add sirloin and cook, flipping once, until medium rare, about 4-5 minutes per side. Let the meat rest for 5-7 minutes before carving into thin slices.
To assemble, top each pita bread with sirloin, tzatziki sauce, tomatoes and red onion. Serve immediately.