Whisk the warm milk with the yeast, sugar, egg, and melted butter in a 4-quart / 3.8 L bowl. Add the flour and salt and stir until a shaggy dough forms.

Knead on a lightly floured counter for 8 to 10 minutes until the dough is smooth and slightly tacky but no longer sticks to your hands. Place it in a lightly oiled bowl, cover, and let it rise in a warm spot for 60 minutes until doubled.

Mash the softened butter with the sugar, flour, and cardamom in a small bowl until it forms a thick, spreadable paste. Use softened butter, never melted, or it will run out of the buns while they bake. Stir the saffron into the hot milk and set it aside to steep.

Line two 33 x 46 cm / 13 x 18 inch baking sheets with parchment. Punch down the dough and divide it into 8 equal pieces, about 110 g each. Keep the pieces you aren't working on covered.

Roll one piece on a lightly floured counter into a rectangle about 35 x 18 cm / 14 x 7 inches, thin enough to faintly see the counter through. Spread about 2 tablespoons of filling over it, leaving a 1 cm border, then roll it up tightly from the long edge into a rope.

Coil the rope into a snug spiral, tuck the outer end underneath, and set it on a sheet. Repeat with the remaining pieces, placing 4 buns per sheet with 5 cm / 2 inches between them. Cover loosely and let rise for 30 minutes until puffy. While they rise, heat the oven to 180 C / 350 F with racks in the upper and lower thirds.

Whisk the egg yolk into the saffron milk. Brush it over the buns and sprinkle with the poppy seeds.

Bake for 22 minutes, swapping the sheets' positions after 11 minutes, until the tops are deep golden brown and the center reads 190 F / 88 C on an instant-read thermometer. Transfer to a rack and serve warm.