Preheat the oven to 425°F (218°C) and place a rack in the middle position.
Place the flatbreads on a baking sheet in a single layer.
Spread about 1/8 cup of prepared pesto over each flatbread, leaving a small border at the edges.
Evenly sprinkle shredded mozzarella over the pesto on each flatbread.
Top each flatbread with shredded chicken, then distribute cherry tomatoes, chopped spinach, sliced red onion, and sliced kalamata olives across the flatbreads.
Scatter crumbled feta over the toppings and drizzle a little additional pesto on top if desired.
Bake on the middle rack for 10–12 minutes, until the cheese is melted and the flatbread edges are crisp.
Remove from the oven, slice each flatbread into 3–4 pieces, and serve immediately.