Warm the milk in a small saucepan to about 110 F — it should feel like a warm bath, not hot. Pour into the bowl of a stand mixer, sprinkle over the yeast and 1 tablespoon of the granulated sugar, and let sit for 8 minutes until foamy on top.
Crush the whole green cardamom pods in a mortar and pestle, discard the papery green husks, and grind the dark seeds to a coarse fragrant powder. You will need 2 tablespoons for the dough and 1 tablespoon for the filling.
To the yeast mixture add the remaining granulated sugar for the dough (about 7 tablespoons), the salt, 2 tablespoons of the crushed cardamom, the egg, and 4 cups of the flour. Mix with the dough hook on low for 3 minutes until a shaggy dough forms.
Add the 6 tablespoons of softened butter one tablespoon at a time, letting each be absorbed before adding the next. Knead on medium speed for 8 to 10 minutes, adding the remaining 0.25 cup flour only if the dough clings to the bottom of the bowl. You want a smooth, glossy, slightly tacky dough that pulls away from the sides.
Cover the bowl with a clean kitchen towel and let rise in a warm spot for 60 minutes, or until nearly doubled in size.
Meanwhile, make the filling. In a small bowl, mash together the 7 tablespoons very soft butter, brown sugar, 1 tablespoon crushed cardamom, and a pinch of salt with a fork until it forms a smooth, spreadable paste.
Turn the risen dough onto a lightly floured surface and roll into a rectangle about 20 inches wide by 12 inches tall, with a long edge facing you.
Spread the cardamom butter evenly over the entire rectangle using an offset spatula, right to the edges.
Fold the dough in thirds like a business letter: bring the bottom third up over the middle, then fold the top third down over that. You now have a long strip about 20 inches wide by 4 inches tall with layers of filling inside. Gently roll it once with the pin to seal.
With a sharp knife or bench scraper, cut the strip crosswise into 14 equal strips, each about 1.5 inches wide.
Shape each bun: hold one strip and give it 3 to 4 twists along its length. Wrap the twisted strip around two fingers of your other hand twice, then tuck the loose end up through the middle and press it under the bun. This is the classic kardemummaknut. Place on a parchment-lined baking sheet, spacing 7 buns per tray.
Cover the trays loosely with kitchen towels and let the buns proof for 45 minutes, until visibly puffy and pillowy.
About 15 minutes before baking, heat the oven to 425 F with racks in the upper and lower thirds.
Brush each risen bun generously with beaten egg, then sprinkle heavily with Swedish pearl sugar, letting it settle into the crevices of the knots.
Bake for 10 to 12 minutes, rotating and swapping the trays halfway through, until the buns are deep golden brown and the exposed filling is bubbling.
While the buns bake, make the finishing syrup: combine 3 tablespoons granulated sugar and 3 tablespoons water in a small saucepan, bring to a simmer, and stir just until the sugar dissolves. Remove from the heat.
The moment the buns come out of the oven, brush each one with the warm syrup for shine and to keep them moist. Let cool on the trays for 10 minutes, then serve warm with hot coffee or black tea for a proper Swedish fika.
