Place the flank steak in a large sealable bag or dish. Whisk together extra virgin olive oil, fresh lemon juice, red wine vinegar, minced garlic, dried oregano, salt, and black pepper. Pour over steak, seal, and refrigerate for 1 to 4 hours. Bring to room temperature 20 minutes before cooking.

Grate the English cucumber and squeeze hard inside a kitchen towel over the sink to remove excess moisture. In a bowl, combine squeezed cucumber, Greek yogurt, grated garlic, chopped dill, lemon juice, olive oil, salt, and pepper. Cover and chill in refrigerator.

Heat a cast-iron skillet or grill pan over medium-high heat until smoking. Remove steak from marinade, discarding liquid. Sear steak for 5 to 7 minutes per side for medium-rare ($135°F). Transfer to a cutting board and let rest for 10 minutes. Slice thinly across the muscle grain at a 45-degree angle.

Warm whole-wheat flatbreads in a dry skillet over medium heat for 30 seconds per side. Spread a generous tablespoon of cool tzatziki down the center of each wrap. Layer with sliced flank steak, cherry tomatoes, cucumbers, and red onion slices. Garnish with chopped herbs and crumbled feta cheese. Roll tightly and serve immediately.