In the bowl of a stand mixer, mix milk and sugar, and sprinkle yeast over it, let it sit for a few minutes or until foamy.
In a separate bowl, whisk flour, cardamom and salt.
Using a paddle attachment, with the mixer on low speed, stir in egg, vanilla and approx 1/4 of flour mixture into the milk and yeast mixture and mix until fully incorporated. With a mixer on low speed, add in softened and cubed butter, piece by piece until fully incorporated, slowly add rest of the flour and stir until all of the flour is moistened.
Switch to the dough hook, and knead for 8-10 minutes or until the dough is smooth, pliable, and stretchy, but still soft and not overly sticky. You may need to add more flour, a tablespoon at the time – you want the mixture to end up a bit sticky, but not so much that it sticks to your finger if you poke it. It is better not to add too much flour as this will result in dry buns.
Transfer the dough to a lightly oiled bowl, cover the bowl with plastic wrap and let the dough rise in a warm spot until doubled in size, about 1-1 1/2 hour.
For the filling, using an electric mixer, mix butter, sugar, and ground cardamom on medium-low until creamy and smooth.
Deflate the dough by punching it down and place on a floured surface. Roll out dough into a 40 x 50 cm (16 x 20 inch) rectangle.
Using a spatula, spread the cardamom butter all over the dough, then fold the dough in half (you will end up with a 20 x 50 cm (8 x 20-inch) rectangle).
Line two baking trays with baking paper.
Using a sharp knife or a pizza cutter, cut the dough in half, then repeat this process until you have used all of it. You should have about 16 equal strips. Now, cut each strip down the middle, but leave it attached on top, about 1.5 cm. Carefully take one strip and wrap it around two of your fingers twice to create a roll then loop the remaining length around the top twice, tucking the end underneath (How to make knots- link for a video is in tips and trick section).
Place each bun on the baking sheet and leave to prove under a dish towel for another 30 minutes.
While proofing, preheat oven to 190°C/375°F at least 30 minutes before baking.
For the syrup, while rolls are proofing, heat water and sugar in a small saucepan on high until sugar has dissolved completely, add in vanilla and cardamom and set aside.
Brush each bun lightly with the beaten egg, sprinkle with sugar pearls and bake in the preheated oven. Bake for 15-20 minutes or until golden brown, (depending on your oven, buns can be done within 12-25 minutes so watch them as they bake: they can go dark very quickly and you may also need to rotate the pan if they are not baking evenly.)
After you take the buns out of the oven, immediately brush tops with prepared syrup.