In a small saucepan, gently heat the milk with the butter and the cardamom. Don't simmer or boil the mixture, just gently heat the milk to melt the butter and steep the cardamom in the warm liquid. This will help to bring out a stronger cardamom flavor in the finished buns.
Transfer the warm mixture to a bowl to cool down to about 98℉ / 36℃ before moving on to the next step.
Now, add the cooled milk mixture to a stand mixer with a dough hook attachment. Add in the yeast and sugar, and stir them in to dissolve. If using active dry yeast, let the mixture sit for 5 minutes to bloom the yeast before adding in the other ingredients.
With the machine mixing on low speed, gradually add in the flour and salt. The dough should pull away from the sides of the bowl once all of the flour is incorporated. If the dough is too wet, add in 1 tablespoon of flour at a time until the dough reaches the right consistency. Knead the dough on medium speed for 10 minutes, or until it can pass the windowpane test, which is where you stretch a small piece of dough and if it doesn't break and you can see light through it, then the gluten has properly developed.
Put the dough in a large bowl and cover it with plastic wrap or a damp towel and set it in a warm spot to rise for 30-45 minutes, or doubled in size.
Once doubled in size, punch down the dough and then divide it into 12 equal pieces. You can eyeball the dough sizes, or you can weigh your dough in grams and then divide that number by 12. That number is the amount of grams each dough ball should weigh. Cover the balls of dough with plastic wrap while you shape them, so that the ones you aren't working on don't dry out.
Form each piece into a bun by pulling in all sides towards the middle and pinching them together. Then put the seam side down on your work surface and roll the ball of dough under your hand in a circular motion to seal the seam. Place the formed buns on a parchment-lined baking sheet spaced a couple of inches apart. They fit best arranged 3 x 3. Cover them with plastic wrap and let them rise for 30 minutes, or until doubled in size.
Once doubled in size, brush them with the egg wash and bake them in a 425℉ / 220℃ preheated oven for 15-20 minutes, or until golden brown. You may want to swap the baking sheets halfway through baking for more even coloring. When finished baking, remove them to a rack to cool completely before filling them. These will take about 1-1½ hours to cool.