Preheat the oven to 390°F/200°C in conventional mode (top and bottom heat). Wash the cherry tomatoes and cut them in half.
In a medium heat pan, sauté finely minced garlic and red onion with olive oil until translucent.
Then, add balsamic vinegar, honey, thyme, oregano, salt, and pepper. Let it cook for a few minutes. Add the cherry tomatoes to the pan and cook for a few more minutes (until they are slightly wrinkled), stirring often. Strain the cooking juice (which you keep) and let the contents of the pan cool aside.
In a tart pan (or directly on a baking sheet lined with parchment paper), place a sheet of parchment paper and your cherry tomatoes. Add 1 tablespoon of cooking juice, brush your puff pastry with pesto, then spread it over the cherry tomatoes, tucking the edges of the pastry.
Make a few holes in the puff pastry with a fork, place it in the middle of the oven, and bake for 20 to 30 minutes, until the pastry is golden and crispy.
Take it out of the oven, let it cool for 5 minutes before turning the tarte tatin onto a plate, and adding chunks of burrata, a few fresh basil leaves. Enjoy!