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 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!)
Combine the egg, egg yolk, sugar, vanilla extract, 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 and softened butter, stirring with a wooden spoon. Once the dough starts to come together, you can start kneading.
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, 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.
For the filling, combine softened butter, brown sugar, cinnamon, and salt and stir together until you get a smooth paste.
Once your dough has doubled in size, roll it out into a rectangle that is about a quarter of an inch (0.5cm) thick.
Spread your filling evenly on top.
Roll it up into a log along the longer edge of the rectangle.
Cut the log in half.
Place the two logs (seam-side down!) on a baking sheet lined with parchment paper. Begin to create the braid pattern by cutting the log with a pair of scissors at a bit of an angle, about an inch from the previous cut. Do not cut all the way through! Pull the piece to the side, alternating each time.
At the end you should have a log that looks like this!
Cover and leave the swirled loafs to rise for another 45 minutes, or until the dough has almost doubled in size and passed the poke test (when poked, the dough springs back slightly, but leaves an indentation). In the meantime, preheat your oven to 350 degrees F (175 degrees C).
Once your dough is ready, brush with an egg wash. Bake for about 20 minutes - make sure to check the center for doneness and add more time if needed!