Preheat the oven to 180℃. Prepare a baking sheet with baking paper.
Roll out the pastry dough into the shape of a circle.
In a small pot cook the blueberries with brown sugar, vanilla syrup, and butter until they are very soft but not mashed. Add cornstarch and stir well. Let cook on medium-low heat for another 3 minutes and stir continuously. If you are using frozen blueberries, defrost them in warm water and get rid of the extra water before cooking.
Add the mixture right into the middle of the puff pastry. Leave a few centimetres from the sides. Fold the sides in so the galette has the shape of a circle. Beat the egg and use egg wash to brush the sides.
For the crumble: mix butter, sugar and flour with your fingers. After a few minutes, you should end up with a very crumbly dough. Sprinkle the galette with crumble.
Sprinkle the edges with brown sugar and bake for 20-25 minutes. Take the galette out of the oven and let it cool. You can serve the galette warm or cold, but I recommend serving it with a scoop of vanilla ice cream.