Warm the milk gently until just lukewarm, then stir in the yeast to dissolve fully. Combine the sugar, cardamom, vanilla, and about two-thirds of the flour with the milk mixture. Knead by hand or with a stand mixer fitted with a dough hook, gradually adding the remaining flour until the dough is smooth and workable but not overly sticky. Incorporate the softened butter and continue kneading for roughly ten minutes until the dough is elastic. Cover and allow to rise until doubled in volume, approximately one hour.

To prepare the custard, heat the milk together with half the sugar in a small saucepan. Slice open the vanilla pod, scrape out the seeds, and add both seeds and pod to the milk. In a bowl, whisk the remaining sugar with the corn starch and egg yolks until silky. Once the milk mixture reaches a gentle bubble, pour half into the egg mixture in a steady stream, whisking constantly. Return everything to the saucepan and cook over medium heat, whisking continuously for about two minutes until the custard thickens notably. Remove from heat and transfer to a clean bowl, covering directly with plastic wrap to prevent a skin from forming. Leave to cool while the dough completes its rise.

When the dough is ready, turn it onto a floured surface and shape into a long cylinder. Slice into 12 to 15 equal portions and roll each into a smooth ball. Arrange on baking trays lined with parchment, ensuring sufficient spacing. Press a deep well into the center of each bun using your fingers or the back of a spoon. Pipe or spoon 1–2 tablespoons of custard into each indentation. Cover loosely and let the buns rise once more for about 30 minutes.

Preheat the oven to 410°F (210°C) with top and bottom heat. Beat the egg and lightly brush each bun with the wash. Bake on the middle rack for 10–14 minutes, until golden brown and cooked through.

Mix the powdered sugar gradually with water until a thick icing forms. Pour the shredded coconut onto a shallow plate. When the buns have cooled completely, spread the icing generously around the custard centers, then dip each bun top into the coconut, pressing lightly to ensure full coverage.