Preheat a skillet or griddle over medium heat.
Trim and slice 1 lb chicken breast into thin strips and place in a medium bowl.
Add 2 tbsp olive oil, 1 tbsp lemon juice, 1 minced garlic clove, 1/2 tsp cumin, 1/4 tsp paprika, 1 tsp dried oregano, 1/2 tsp rosemary, and salt and pepper to the chicken; toss until evenly coated.
Cook the chicken on the preheated skillet for about 5–7 minutes, stirring occasionally, until cooked through and no longer pink.
While the chicken cooks, prepare the tzatziki by combining 1/2 cup plain Greek yogurt, zest of 1/2 lemon and 1 tsp lemon juice, 1 tbsp olive oil, 1 minced garlic clove, 1 tbsp chopped fresh dill, and 1/4 tsp salt and 1/4 tsp pepper; whisk until smooth (or pulse in a food processor for creamier texture). Chill briefly if desired.
Warm the flatbreads or pitas if desired. Divide the cooked chicken among 2 flatbreads and top with sliced green pepper, sliced onion, and pickles or cucumbers.
Drizzle tzatziki sauce over the fillings, fold the flatbreads, and serve immediately.