Heat up the milk, either in the microwave or in a saucepan, until it is "fingervarm" (finger warm), meaning it should feel warm to the touch but not hot. You don't want to kill the yeast! Add the yeast and a bit of the sugar from the recipe and stir to dissolve. Let it rest 5-10 minutes until it begins to froth and bubble a little bit. (If your yeast doesn't do anything, it may be expired. Check the expiration date on the package and start over with new yeast!)
While the yeast sits, grate a few apples and squeeze out the liquid. If weighing the grated apple, be sure to weigh it after you squeeze out the liquid.
Combine the egg yolk, grated apple, remaining sugar, vanilla extract, cinnamon, cardamom, and salt in a large bowl. Mix well.
Add the milk and yeast mixture to the egg mixture and whisk to combine.
Slowly add the flour, stirring with a wooden spoon. Once the dough starts to come together, you can start kneading. Note: you may not need to add all of the flour, which is why we recommend adding it slowly and gauging how dry the dough appears. You can always add more flour as you knead, but it's very difficult to get moisture back into the dough once it is too dry.
Add the softened butter and knead until smooth and elastic, probably around 5-10 minutes. Add a bit of flour if the dough sticks to your hands or the surface. If you prefer not to get your hands messy and covered in butter, feel free to use a stand mixer for this step. The texture of your dough before its first rise should be soft and smooth, not sticky and not too dry.
Cover and let your dough rise for about 45 minutes, or until doubled in size. You can use the "poke" test to check whether your dough is done rising; if you poke the dough and it leaves an indentation and doesn't spring back immediately, it should be ready!
For the filling, combine softened butter, brown sugar, cinnamon, and salt and stir together until you get a smooth paste.
While the dough rises, chop up 1.5-2 medium sized apples into small chunks. If you're not sure about the size of your apples, start with less! You can always chop up more apple to add before you roll up the dough. Cutting them up smaller ensures they soften when they bake.
Once your dough has doubled in size, roll it out into a rectangle that is about a quarter of an inch (0.5cm) thick (should be about 11" by 16" or 12" by 17" in size).
Spread your cinnamon and butter filling evenly on top.
Sprinkle over the apples in an even layer. You may have some apple leftover!
Roll up the dough into a log. You'll want to roll from the shorter edge of the rectangle, so that your buns have more swirls.
Cut the log into 1-1.5 inch (3cm) thick pieces; we cut ours into 12 pieces.
Place the buns (plenty of space in between!) on two baking sheets lined with parchment paper or a silicone baking mat.
Cover the buns loosely with cling film or a clean dishtowel and let them rise another 45 minutes. In the meantime, preheat your oven to 350 degrees F (175 degrees C).
You can use the "poke" test to check whether your dough is done rising; if you poke the dough and it leaves an indentation and doesn't spring back immediately, it should be ready! The apples may also "weep" a bit of liquid as they are sitting; we haven't had this cause any issues, but there may be a little bit of liquid around the buns!
Make your egg wash (1 egg whisked together with a splash of water until there are no more gloopy streaks of egg white), then brush over the buns.
Bake for about 15 minutes, perhaps a few minutes longer if not using a convection oven. Once they are done, transfer to a wire rack to cool for at least 20 minutes.
Combine powdered sugar, a bit of cinnamon, and water/milk until you reach your desired icing consistency. Remember, a little bit of liquid goes a long way, so start with a cup of powdered sugar and a tiny bit of water or milk! Pour or drizzle over the kanelsnegle and enjoy!