Pour milk into the bowl of a stand mixer. Sprinkle yeast on top, followed by 1 ⅓ cups flour and mix with a wooden spoon. Allow to stand until foamy, about 20 minutes.

Add flour, softened butter, powdered sugar, egg, cardamom, and salt; knead with dough hook (or by hand) until a soft, smooth, elastic dough forms, about 5 to 6 minutes.

Transfer dough to a lightly floured surface and shape into a ball. Divide into 12 equal portions. Using flour as necessary, roll dough under a cupped hand to form perfectly smooth balls.

Place dough balls evenly on a parchment-lined baking sheet, leaving space in between, and slightly push down. Dust tops with flour, and cover with a clean kitchen towel. Let rise in a warm spot until doubled in size, about 2 hours.

Preheat the oven to 425 degrees F (220 degrees C).

Stir together egg and water in a small bowl and brush rolls all over with egg wash.

Bake in the preheated oven until lightly golden, about 12 minutes. Transfer buns onto a wire rack and let cool completely.

Combine almond paste, powdered sugar, milk, and cardamom in a food processor and blend almond filling until smooth.