Preheat your oven to 400°F (200°C).
In a mixing bowl, combine the shredded chicken and basil pesto, mixing until the chicken is well coated.
Place the flatbreads on a baking sheet. Brush each flatbread lightly with olive oil.
Evenly distribute the pesto chicken mixture over each flatbread.
Sprinkle the shredded mozzarella cheese on top of the chicken.
Add the halved cherry tomatoes and sliced red onion over the cheese. Season with salt and pepper to taste.
Bake in the preheated oven for 12-15 minutes, or until the cheese is melted and bubbly, and the edges of the flatbread are golden brown.
Remove from the oven and let cool for a couple of minutes. Garnish with fresh basil leaves if desired.
Slice into pieces and serve warm.