If using active dry yeast, heat the milk in the microwave or on the stovetop until it is “fingervarm” (finger warm), meaning it should feel warm to the touch but not hot. Add your yeast, stir, and leave it for 5-10 minutes until the mixture starts to bubble up. If using instant yeast, simply add the yeast to the milk – there is no need to dissolve it.
Combine the egg, egg yolk, sugar, vanilla extract, salt, and pumpkin puree in a large bowl. Mix well.
Once the yeast mixture has bubbled up a bit, add it to the egg mixture, or if using instant yeast, you can add it immediately. Mix well again.
Then add the softened butter and flour. Mix with a wooden spoon until the dough starts to come together. Tip it out onto a floured surface and knead until smooth and elastic (about 10 minutes). Add more flour if needed – the dough should be soft and smooth, not dry and not sticking to the surface. If using a stand mixer, allow it to knead the dough together for about 10 minutes.
Cover and let your dough rise for about 45 minutes, or until doubled in size.
For the filling, combine softened butter, brown sugar, cinnamon, pumpkin spice, 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 and roll it up 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. Here is where you can customize the recipe: place the pieces (swirl side up!) close together in a buttered baking dish for gooey, American-style cinnamon buns, or separate them (plenty of space in between!) on two baking sheets lined with parchment paper for more of a crispy, Danish-style cinnamon snail.
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).
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 and bake for about 15 minutes.
Once they are done, let them cool, ice the buns if you want to (make icing by combining powdered sugar with a tiny amount of water and mixing until you get a consistency that you can drizzle over the buns), and then enjoy!