To clarify the stock: Beat 1 cup stock in a bowl with the egg whites.
Bring remaining stock to a boil in a saucepan.
Dribble the hot stock into the egg-white mixture, beating constantly with a whisk.
Return all stock to the saucepan and bring to a simmer over moderate heat, stirring gently with the whisk at all times.
When stock reaches a simmer, stop stirring and reduce heat to absolute minimum; liquid must just shiver for 20 minutes.
Meanwhile, set a sieve or colander over a bowl and line with several layers of damp cheesecloth or paper towels.
Set the saucepan of stock next to the bowl and gently ladle broth and egg whites into the sieve, keeping whites intact as much as possible.
The egg whites and particles from the stock will remain behind in the strainer.
The resulting stock will be transparent and golden.
Put stock in a clean pan and bring to a boil.
Drop in the asparagus and cook until just tender, less than 5 minutes.
Ladle into serving bowls, and garnish with chopped fresh dill.