Steep fennel, anise, and Parmigiano-Reggiano rind in chicken stock for 45 minutes. Strain and pour broth into a bowl.
Bring a pot of salted water to a boil. Cook the asparagus for 3-4 minutes.
Remove the cooked asparagus. Place it into a bowl of ice water for a few seconds. Remove and set aside.
Purée one asparagus stalk a food processor until smooth, adding water to thin if necessary.
Slice the other piece of blanched asparagus into bite-sized pieces. Save for garnish.
Bring a pot of water to a boil and reduce to a simmer. Add the vinegar.
Crack the egg into a mug. Slowly pour it into the simmering vinegar water. Cook until the whites are done and the yolk begins to firm.
Remove the egg from the water with a slotted spoon. Place in the bowl of broth.
Drizzle the asparagus purée on top, and add garnish of chopped asparagus.