Gently heat 1 cup of milk, 3 tablespoons of butter, and 1/4 cup of powdered sugar until warm. Stir in 2 teaspoons of instant yeast and 1 teaspoon of freshly ground cardamom, then gradually add 4 cups of flour to form a smooth dough. Knead for 10 minutes, cover, and let rise in a warm place for 1 hour.

Heat 2 cups of milk with half of the 1/2 cup of sugar and a split vanilla pod until bubbling. In a separate bowl, combine remaining sugar, 2 tablespoons of flour, and 2 tablespoons of corn starch. Whisk in 4 egg yolks and slowly combine with hot milk. Cook for 2 minutes until thickened, then cover and cool.

Mix 1/4 cup of softened butter with 1/4 cup of granulated sugar and 1 teaspoon of cinnamon. Roll the dough into a large rectangle on a floured surface.

Spread cinnamon filling over the dough, roll tightly, and slice into 22 equal pieces. Make a thumbprint in each bun and place on a baking tray. Let rise for 30 minutes.

Preheat oven to 445°F (230°C). Brush tops with an egg wash and bake for about 10 minutes until golden brown. Transfer to cooling rack.

Mix 1 cup of powdered sugar with water for icing. Drizzle over cooled buns.