In a medium bowl, combine shrimp, 1 tablespoon olive oil, minced garlic, smoked paprika, dried oregano, salt, black pepper, and lemon juice. Toss to coat and marinate for 10–15 minutes at room temperature.
In a separate small bowl, toss together diced cucumber, Roma tomatoes, and red onion. Set aside.
Heat a cast-iron skillet or grill pan over medium-high heat. Add marinated shrimp in a single layer and sear for 2–3 minutes per side until pink, opaque, and lightly charred. Transfer to a warm plate.
Wipe skillet clean, brush flatbreads lightly with olive oil, and warm for 1–2 minutes per side until soft and toasted.
Spread 1/4 cup tzatziki sauce over each warm flatbread. Top with seared shrimp, spoon the diced vegetable mixture over top, and finish with crumbled feta cheese and chopped fresh parsley. Serve hot with lemon wedges.