In a large bowl, whisk together minced garlic, lemon juice, red wine vinegar, olive oil, Greek yogurt, dried oregano, salt, and pepper. Add the beef flank steak, ensuring it is well coated. Cover and refrigerate for about 1 hour.
Shred the cucumber or chop it in a food processor. Toss with ½ teaspoon of salt and let it sit over a strainer for 20-30 minutes to drain excess liquid. Squeeze to remove moisture.
In a medium bowl, combine drained cucumber, Greek yogurt, lemon juice, dill, minced garlic, and white wine vinegar. Season with salt and pepper, drizzle with olive oil, and refrigerate for at least 30 minutes.
Preheat your oven to 400°F (200°C). Coat a baking dish with oil and arrange marinated beef in a single layer. Cover with parchment paper. Bake for 12 to 18 minutes or until the internal temperature reaches 140°F (60°C). Let it rest for 5 minutes before slicing.
Alternatively, preheat your grill to medium-high heat and grill the beef for 8 to 10 minutes until it reaches 140°F (60°C). Let it rest before slicing.
Serve the sliced beef on pita bread, topped with tzatziki sauce, fresh dill, cubed feta cheese, chopped tomatoes, and thinly sliced red onion.
Notes
For a spicier kick, add chopped jalapeños to the wraps. You can also use chicken or lamb instead of beef if desired.