Prepare Danish pastry cream (kagecreme) at least one hour in advance of preparing the dough; you can also make this the day before! To prepare the dough: 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. Measure out the sugar you will need for the dough in this recipe. If you're using active dry or fresh yeast, add the yeast along with a few teaspoons of the sugar you measured out into the warm milk 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!If you're using instant yeast, you do not usually need to bloom it ahead of time and can add it right into a mixing bowl with all your other ingredients. If in doubt, check the package directions of the yeast you are using! Combine the egg, egg yolk, rest of the sugar, cardamom, vanilla extract, and salt in a large bowl or the bowl of a stand mixer fitted with the dough hook attachment. Mix well.
Add the milk and yeast mixture to the egg mixture and whisk to combine.
Add the flour, stirring with a spoon or running the mixer for about 30 seconds. Once the dough starts to come together, you can start kneading.
After the flour has been added, add the softened butter and knead until smooth and elastic, probably around 10 minutes. Add a bit of flour if the dough sticks to your hands or the surface. If you're using a stand mixer and notice the dough sticking to the sides, be sure to add a little bit more flour as well. The texture of your dough before its first rise should be soft and smooth, not sticky and not too dry. If the dough is feeling greasy in any way from the butter, it has likely not been kneaded enough – keep kneading and adding flour if you think it needs it! Cover and let your dough rise for about 45 minutes, or until doubled in size. You can also use the "poke test" to determine whether the dough is ready (when you poke the dough, it should create an indentation and slowly spring back - if it springs back immediately, it needs more time).
While the dough is rising, make your cinnamon butter filling by combining softened butter, brown sugar, cinnamon, and a pinch of salt in a bowl and mixing until you have a smooth and spreadable paste.
Prepare a springform pan (a 9-inch or 10-inch pan will work for this recipe) by oiling or buttering the sides and lining the bottom with parchment paper.
Once the dough has doubled in size, turn it out onto a clean work surface.
Take approximately one-third of the dough and roll it out into a circle shape. It should be slightly larger than the bottom of your springform pan.
Transfer the dough circle to your pan, pressing it into the corners and making sure it goes up the sides by at least ½" or about 1¼ cm.
Add about 3 tbsp of the cinnamon filling on top of the circle and spread to the edges. If you do need to add a little bit more filling, feel free to do so!
Add a few dollops of the cooled pastry cream and spread all the way to the edges of the circle - we are not the most precise with this amount, but an even thin layer covering all of the cinnamon filling should suffice!
With the remaining two-thirds of the dough, roll it out into a rectangle that is about 10" x 14" (~25cm x 35cm)
Spread the remaining cinnamon filling in an even layer over the top, all the way to the edges.
Spread the remaining cooled pastry cream over the top of the cinnamon filling, in an even layer and all the way to the edges.
It's time to roll up the dough - this might get a bit messy!
Roll from the shorter edge of the rectangle to create rolls with more swirls. Roll as tightly as you can without squishing out the filling.
Divide the dough log into 8 equal pieces; measure by marking the halfway point, then the halfway point of both halves, and then the halfway mark of the quarters.
Slice the dough log into rolls with a sharp knife - some filling may come out, which is okay!
Transfer the rolls to the springform pan, arranging them as evenly as possible. There will be space between them - they will not tightly fill the space - but this is because there needs to be room for the dough to rise!
Let rise for another 45 minutes. The dough should have filled a lot of the extra space and become puffy and airy.
Preheat oven to 350 degrees F (175 degrees C).
Once the dough is done rising, brush the top with an egg wash - crack an egg into a bowl, add about a tablespoon of water, and whisk together, then use a pastry brush to brush this over the top of the rolls.
Bake for 45-50 minutes. Check the internal temperature, and it should reach 88ish C (190 F) when they’re fully baked through in the middle. Cover with foil if they get too dark on the top but are still underbaked in the middle.
Once done, let cool for about 10 minutes before removing the side of the springform pan.
Create a simple icing by combining powdered sugar and a tiny bit of water in a bowl - we usually don't measure this, but make sure to add less water than you think to keep it on the thicker side!
Spread over the top of the cake when it has cooled a little bit more. Slice and enjoy! We keep this in the fridge if not eaten immediately due to the pastry cream.