Bring the water and butter to a boil in a saucepan
Meanwhile, mix the dry ingredients together (flour, cardamom, sugar, and salt) and set aside
Also, crack the eggs into a bowl, and set aside
When the water/butter has come to a boil, reduce the heat and add the dry ingredients all at once
Mix vigorously with a wooden spoon until the dough holds together and is shiny
Remove from the heat
Add the eggs, one at a time, and mix thoroughly. The dough will break up when the eggs are added, but just keep stirring with gusto! The dough will be soft, but should hold together.
Cool for about an hour

Bring water to a slight boil in a 2-quart pot, add 1/4 tsp salt
Put the dough in a Danish dumpling press (it's like a cookie press), press the dough through the press and use a knife to cut the dough off in 1/2" dumplings. If you do not have a bollesprøjte, use two spoons and drop teaspoon-sized dumplings into the water. Only put one layer of dumplings into the water at once so they cook uniformly and don't bump into each other as they cook (I got a little carried away and added too many, as you can see in the photo above.)
Here's the tricky part: Bring the water almost to a boil and then at the point where the water is about to boil add 3-4 ice cubes to cool the water.   You will do this three times. After the third time, use a slotted spoon to gently lift the dumplings out of the water Drop the dumplings in a bowl of ice water to cool quickly. While they are cooling I add the next batch of dumplings to the water Remove the first batch out of the ice water and drain in a colander Cook the rest of the dumplings the same manner
You can refrigerate the dumplings until needed, and heat them gently in a warm oven before serving. (I wouldn't keep them refrigerated longer than a couple of days.)

Bring the water almost to a boil and then at the point where the water is about to boil add 3-4 ice cubes to cool the water.   You will do this three times.
After the third time, use a slotted spoon to gently lift the dumplings out of the water
Drop the dumplings in a bowl of ice water to cool quickly. While they are cooling I add the next batch of dumplings to the water
Remove the first batch out of the ice water and drain in a colander
Cook the rest of the dumplings the same manner