Heat the milk, butter, and powdered sugar in a saucepan. Cool until just a bit warmer than lukewarm.
Pour the milk mixture into a mixing bowl and add the cardamom and yeast. Stir until dissolved and then slowly add the flour. You may need more or less flour. Add enough so that you can knead the dough into a nice ball without being too sticky. Knead for about 10 minutes, then cover and let rise in a warm place until double in size (about an hour, possibly longer if using dry yeast).
Pumpkin filling
Heat the milk, pumpkin and granulated sugar in a saucepan.
Mix together the brown sugar, flour, corn starch, and spices in a bowl. Add the egg yolks and egg and whisk together until smooth.
Once the pumpkin milk mixture starts to bubble pour about half of it into the bowl with the egg mixture, whisking quickly the entire time. Then return the saucepan to the heat and pour the egg and pumpkin milk mixture from the bowl back into the saucepan, whisking the whole time. Let the pumpkin custard cook for 2 minutes, whisking the entire time. By the end it should be quite thick and harder to whisk.
After 2 minutes remove the pumpkin custard from the heat and pour into a clean bowl. Cover with plastic wrap, placing it directly touching the top of the custard. Let the pumpkin custard cool while the dough finishes rising.
Cinnamon filling
Mix together the butter, brown sugar, cinnamon, and cloves. If the butter is still hard you can pop it in the microwave for a few seconds so it's easy to spread.
Roll out the dough into a large square. Spread the cinnamon butter mixture across all of the square, leaving a few centimeters (2-3 inches) at the top AND bottom of the square.
Roll the rectangle from the bottom up into a long cylinder. Use a knife to cut (~1 inch) slices out of the cylinder. This should make about 22 buns. Set the buns spaced apart on baking sheets.
Press down a large well in each bun. Be sure to press the dough all the way down and make a bigger well than you'll actually need, as the dough will puff back up a bit.
Use a piping bag or spoon to place pumpkin filling in the well of each bun. Cover and let rise again for 30 minutes.
Baking & Decoration
Preheat oven to 445°F (230°C). Whisk egg and carefully brush each bun with the egg wash (you can also skip this step). Bake the pumpkin cinnamon buns for about 10 minutes, until golden brown.
Cool buns on a wire rack. When the buns are fully cooled, mix together powdered sugar and water until you get a thick glaze. You can also add a couple spoons of the leftover egg whites (from the vanilla custard) to make the glaze extra white.
Add the glaze to a small plastic bag and cut the tip off. Drizzle the buns with the glaze.