Preheat oven to 425 degrees F. Make the shawarma spice mix: In a small bowl, combine the paprika, coriander, cumin, cinnamon, and cayenne pepper. If using a store-bought blend, skip this step.
Cut green tops off carrots and set aside for the yogurt. Thoroughly scrub outside of carrots to remove any dirt. Cut each carrot crosswise into approximately 1 to 1½-inch pieces. On a rimmed baking sheet, toss carrots with 3 tablespoons olive oil and the shawarma spice mix. Roast carrots, tossing halfway through, until golden and caramelized in places, about 30 minutes.
Meanwhile, remove leaves from carrot tops and wash and dry thoroughly. Measure ½ cup of the greens and then place them in a food processor (save any remaining greens for another recipe). Add the yogurt, 2 tablespoons olive oil, lemon juice, garlic, and ¼ cup water. Process until combined and smooth, about 30 seconds. Season with salt and pepper.
In a medium bowl, combine cucumber, romaine, tomatoes, onion, and mint.
Top pitas with roasted carrots, chopped salad, and carrot top sauce.