Heat the oil in a large skillet over medium-high heat. Season the chicken with the cayenne, ½ teaspoon salt, and ¼ teaspoon black pepper. Cook until an instant read thermometer inserted in the center of each piece registers 165º F, 6 to 7 minutes per side. Cut into bite-size pieces.
Meanwhile, mix together the yogurt, cucumber, mint, garlic, ½ teaspoon salt, and ¼ teaspoon black pepper in a medium bowl.
Top the flat breads with the romaine, chicken, tomatoes, and tzatziki.
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