For the pastry: Begin by quickly making the puff pastry. Butter a springform pan or a pastry ring, and lightly cover with brown sugar for a caramelized taste. White sugar also works. Press the pastry firmly into the corners of the mold and trim excess. Place in the freezer while you continue.
For the filling: Gently heat the milk with vanilla sugar and the scraped vanilla pod on low heat. Remember to whisk occasionally as the milk can start to burn at the bottom of the pot. In another bowl, whisk the eggs with sugar and cornstarch. You can use an immersion blender for a perfectly smooth result, though it’s not mandatory. Once the milk is simmering, pour it into the egg mixture while whisking. Transfer back into the pot and heat on low, whisking until the first bubbles appear and the mixture thickens like pastry cream. Transfer to a large bowl, stir in the heavy cream, cover with plastic wrap, and let rest for 1 hour at room temperature.
For baking: Fill the frozen pastry shell with the flan filling, leaving a half-centimeter gap as the filling will puff up in the oven. Smooth with a spatula and bake for 25 minutes at 200°C (fan-assisted), then an additional 5 minutes at 230°C. For a glossy finish, brush with syrup (20 g sugar/20 g water) or glaze once the flan is completely cool.