Wash the asparagus spears and cut the green parts into rounds, keeping the tips as individual inch-long pieces and setting them aside.
Peel and dice the potatoes.
Mince the onion and sauté it in the butter with the asparagus rounds, potato, and mint. Check seasoning, add some of the broth, and cook the mixture for 20 minutes or so.
During this time boil the asparagus tips for a few minutes in a little more of the broth.
When the 20 minutes are up, blend the mixture containing the potatoes and stir it into the main broth pot together with the broth containing the asparagus tips, bring everything to a boil, add the stelline, and cook until they are done.