Preheat oven to 400 degrees. Line a baking sheet with a silpat or parchment paper.
Slightly roll out puff pastry on a lightly floured surface so it's about a 1/4 inch longer on each side. Transfer to prepared baking sheet. Use a fork to prick the entire surface, leaving a 1/4 inch border on all sides.
Brush with pesto. Layer tomatoes on puff pastry. Drizzle with olive oil. Season with salt and pepper. Place in the oven and bake until light golden brown, about 20 minutes.
Remove from oven and dot with burrata cheese (just tear with hands). Place back in the oven for another 2-3 minutes until cheese is warm. Cut into 12 pieces and serve.
(Optional: Sprinkle with parmesan cheese and chopped parsley)