Heat the milk, in the microwave or a saucepan, until it is “fingervarm” (finger warm), meaning it should feel warm to the touch but not hot. Add your yeast and one tablespoon of the sugar, stir, and leave it for 5-10 minutes until the mixture starts to bubble up, meaning your yeast is alive and not expired.
Grate the apple; place the apple into a clean cloth or sturdy paper towel; squeeze liquid out of the apple over the sink - try to get as much liquid out as possible!
Add the remaining two tablespoons of sugar, egg yolk, grated apple, cinnamon, and salt to a large bowl (or the bowl of a stand mixer) and mix well. Once the yeast mixture has bubbled up a bit, add it to the egg mixture. Mix well.
Slowly add the flour and softened butter, mixing with a wooden spoon until the dough starts to come together. You can also do this in a stand mixer with the dough hook attachment.
Knead with your hands until smooth and elastic, about 10 minutes. Add more flour if the dough is too sticky; it should be soft and smooth, not too sticky nor too dry. Use a stand mixer to knead the dough if you would prefer!
Cover the dough with cling film or a clean dishtowel and place in the fridge overnight for 8 hours, or let rise at room temperature (if room temp is very cold wherever you are, put it in a warm place, i.e. under a light) for about 1 hour or until doubled in size.
Once your dough has doubled in size, prepare a baking sheet lined with parchment paper or a silicone mat. To shape the rolls properly, you should first divide your dough into 12 equal portions.
Take a portion, pinch together into a ball shape, then cup it in your hand with the top of the roll touching your palm. Place your hand on an unfloured surface with the seam touching the table. Move your cupped hand in quick, tight circles – this will cause the seam to come together and create tension on the surface of the dough, meaning it will rise upwards in the oven, instead of flattening.
Repeat with the 12 remaining portions of dough!
Cover and let rise for another 45 minutes to an hour. In the meantime, preheat your oven to 375 degrees F (190 degrees C)
Once the buns have risen, brush with egg wash.
Bake for about 15 minutes until golden brown. Let cool on a wire rack, then enjoy!