Mix the butter and milk together in a small saucepan.
Heat over medium-low heat, frequently stirring until the butter is melted or until the mixture reaches 110 degrees F. The mixture should be warm, but not hot to the touch.
In a separate large mixing bowl, briefly whisk the sugar, flour, yeast, 2 teaspoons of the ground cardamom, and the sea salt together until combined.
Pour the warm milk mixture into the dry ingredients, stirring until the dough begins to take shape.
Place the dough onto a floured surface, then knead until the dough is soft and smooth, about 7-10 minutes.
Shape the dough into a ball, then set it into a greased bowl and cover with plastic wrap or a damp towel.
Allow the dough to rise in a warm space until it has doubled in size, about 1 hour.
Mix the 1/2 cup of butter, light brown sugar, cinnamon, the remaining cardamom, and the vanilla together in a separate mixing bowl until evenly combined.
Once the dough has about doubled in size, turn it out onto a large floured surface.
Using a rolling pin, roll the dough into a 22x15-inch rectangle.
Evenly and gently spread the filling mixture across the entire surface of the dough.
Fold the left third of the dough in towards the center, then fold the right third of the dough over that towards the center, similar to folding a business letter.
Carefully run the rolling pin over the dough, pressing out any big bubbles.
Roll the dough out again so it molds into a 15x8-inch rectangle.
Using a pastry cutter, pizza cutter, or knife, slice the dough evenly into 1x8-inch long strips.
Taking hold of one strip, twist it multiple times, carefully stretching until it almost doubles in length. Grab one end of the twisted strip, then loosely wrap the dough around two fingers twice, similar to a bandage.
You should have a round circle of dough around your fingers and a remaining strip. At about a 90 degree angle, loop the remaining strip around the top and bottom of the round circle, forming a knot. Tuck the loose end of the strip in at the bottom.
Move the formed dough bun to a large parchment-covered baking sheet. Repeat with the remaining dough strips.
Using a clean kitchen towel, loosely cover the dough buns for 45-60 minutes, allowing them to rise.
Preheat the oven to 375 degrees F.
Brush each dough bun with the egg wash, then sprinkle with the pearled sugar.
Bake until golden brown, about 15-18 minutes.
Move the buns to a wire baking rack, letting them cool for up to 5 minutes. Serve warm!