In the bowl of a stand mixer, combine all pre-fermented dough ingredients. Place in a warm place for 1 hour (I use the oven with the lights turned on).
When it has doubled in volume, is bubbly, and has an acidic smell, add the bread flour, milk, sugar, cardamom, and salt and place it in a stand mixer fitted with a hook attachment. Knead on medium for 3 minutes or until the dough looks more homogenous and pulls easily from the sides of the bowl but still sticks to your hands.
Then add the butter one tablespoon at a time and knead until it’s completely incorporated. When all the butter is in, knead the dough for 10-12 minutes on medium until it pulls away from the sides of the bowl and you can stretch a small piece of it into a square, creating a film thin enough for light to pass through (windowpane test). The dough won’t stick to your hands anymore and it will look hydrated, silky, and smooth.
Cover the bowl with plastic wrap and leave it at room temperature to proof for 1.5 to 2 hours, depending on how warm it is in your kitchen, or until it doubles in volume.
Punch down the dough and remove it onto a working surface. If it sticks, dust it with a little bit of flour. Divide the dough into 10 pieces (approximately 60g each) and form balls. Place them on two lined parchment paper baking sheets, leaving at least 2” (5cm) in between. Cover them with plastic wrap and leave them at room temperature to proof for 1 hour or until doubled in volume.
Meanwhile, preheat the oven to 390F (200C).
Brush the buns with the egg wash (optional) and bake for 10-12 minutes or until puffed up and golden brown.
Remove from the oven onto a cooling rack to cool down completely before filling with the almond paste and frosting.