Preheat oven to 400 degrees Fahrenheit (200 degrees Celsius).
Place naan breads on a parchment-lined baking sheet.
Spread 2 tablespoons of tzatziki sauce over each naan piece, leaving a small border.
Distribute the cooked chicken, red onion, cherry tomatoes, and kalamata olives evenly over the sauce.
Top each pizza with shredded mozzarella and a sprinkle of dried oregano.
Bake for 8 to 10 minutes until the cheese is bubbly and the edges are crisp.
Remove from oven and garnish with fresh diced cucumber and chopped dill.