Preheat oven to 400 degrees F. Unwrap puff pastry and lay flat on a baking sheet lined with parchment paper. Lay tomato slices on paper towels to drain.
Spread pesto over puff pastry, leaving about 1/2 inch border around the edges.
Top with tomato slices, arranging them so they are not overlapping, if possible. Sprinkle with salt and pepper to taste.
Bake tart for 18-20 minutes, until pastry is puffed and golden brown at edges. While tart is baking, cut burrata into even pieces so it can be arranged over the tart.
Arrange burrata pieces on the baked tart, and top with fresh basil leaves and salt and pepper, to taste. Serve!
Notes
This tart is best the day it's made, although it will keep in the fridge for a day or two if needed. Don't microwave puff pastry - it loses its puffiness.