Arrange an oven rack in top third of oven. Place a large baking sheet on rack and preheat oven to 400°.
In a medium bowl, mix lamb with 1/4 cup chopped mint, 1 tsp. salt, 1/4 tsp. pepper, the yellow onion, bell pepper, ginger, garlic, cumin, cayenne, and cinnamon. Divide lamb mixture between flatbreads and spread evenly all the way to edges. Arrange breads on hot baking sheet and bake until lamb is cooked through and bread is crisp, 12 to 15 minutes.
Meanwhile, make salad: Thickly slice cucumber, cut slices into quarters, and put in a medium bowl. Add feta, tomatoes, red onion, remaining 1/4 cup chopped mint, remaining 1/4 tsp. each salt and pepper, the vinegar, and oil and toss until well combined.
Cut flatbreads into quarters, top with mint leaves, and serve with feta salad.