To make the dough, heat the milk until bubbles form around the edges. Remove from heat and let cool to 100°F. Sprinkle yeast over and let sit for 5 minutes.
While the milk is heating up, cream the butter, sugar, and cardamom in the bowl of an electric mixer with the paddle attachment.
Pour the milk and yeast mixture into the butter and add flour and salt. Mix until blended.
Switch to the dough hook and knead until smooth, about 5 minutes.
Add the dough to a bowl sprayed with cooking spray. Cover with plastic wrap and let proof in a warm place for 1 hour. Turn out onto a floured surface and divide into two balls.
To make the filling, whisk butter, sugar, cardamom, salt, and vanilla bean paste until smooth. Set aside.
Roll the dough out into a rectangle. Spread the filling evenly over the dough.
Fold the bottom third of the rectangle to the middle and fold the top third over that, like a brochure. Slice into inch pieces.
Stretch a piece of dough out a bit, then tie into a knot and place on a parchment-lined baking sheet. Cover and let rise for another 30 minutes.
When ready to bake, whisk egg and water and brush over the buns. Sprinkle half of the buns with sugar pearls and the other half cardamom sugar.
Bake in a preheated 425° oven for 12 to 15 minutes.