Rinse the asparagus and break off the tough, woody ends.
Cut the tips off the asparagus, slice lengthwise and set aside.
Roughly chop the remaining asparagus.
Slice the white part of the leaks, and the celery stocks, and set aside.
Bring a small saucepan of water to boil, lightly salt, and blanch the asparagus tips until tender crisp.
Remove from the sauce pan with a slotted spoon, run under cold water, and set aside on a dishtowel to dry.
Bring a medium sized pot of water to boil, with about 4 inches deep water.
Lower the water temperature to a simmer, then add a teaspoon of vinegar to the water.
Use a wooden spoon to swirl the water until you have a rapidly moving whirlpool of water in your pan.
Crack one of the eggs into a small bowl or large spoon and immediately slide the egg into the pot of water and continue using the wooden spoon to keep the water swirling.
Once the egg has begun to cook and taken shape, allow it to cook for another three minutes.
Use a slotted spoon to remove the egg and place it in a dish of cold water.
Repeat with the remaining three eggs.
In a sauce pan large enough to hold the broth and asparagus heat the olive oil and then add the chopped asparagus, leaks and celery.
Cook for about 10 minutes over a low flame until the vegetables are tender.
Sprinkle the vegetables with the flour and cook stirring for about a minute.
Gradually add the broth stirring constantly, and simmer for an additional 10 to 15 minutes.
Turn off the heat and using an immersion blender, blend the soup until creamy and smooth.
Taste the soup and add salt and freshly ground black pepper to taste.
If the soup is not completely smooth, pour it through a fine mesh strainer.
At this point returned the soup to the stove and stir in the fresh cream.
When you're ready to serve the soup return the poached eggs to the simmering water and allow to warm up for 30 seconds.
Warm up the soup, and pour into individual soup bowls.
Top each bowl with a poached egg and some of the asparagus tips.
If desired drizzle with olive oil and serve.