In a mixing bowl, combine 1 cup of flour, sugar, cardamom, and salt.
In a small saucepan, warm the milk and melt the butter. Add to the flour mixture along with the beaten egg and yeast.
Beat until smooth. Gradually add the remaining flour to form a soft dough.
Turn onto a floured surface and knead until smooth and elastic, about 6-8 minutes.
Place in a greased bowl, turning once to grease the top. Cover and let rise in a warm place until doubled, about 1 hour.
For the filling, mix brown sugar and cinnamon. Punch down dough and roll into a 12x9-in. rectangle. Spread with softened butter and sprinkle with the sugar mixture.
Roll up jelly-roll style, starting with a long side. Place seam side down on a greased baking sheet; pinch ends together to form a ring.
With scissors, cut from the outside edge two-thirds of the way toward the center of the ring at 1-in. intervals. Separate strips slightly; twist to allow filling to show.
Cover and let rise until doubled, about 30 minutes. Bake at 375°F for 20-25 minutes or until golden brown.
Cool on a wire rack. If desired, drizzle with icing.