Chop the remaining part of the spears into 1-inch pieces and drop into the boiling chicken broth, cover, reduce heat, and simmer 20-30 minutes, or until asparagus is very soft.
Puree the soup in a food processor fitted with a steel blade, or in a blender, until smooth.
Return the puree to the pot, add the reserved asparagus tips, and simmer until they are tender but still firm, 5-8 minutes.
If serving the soup hot, stir in the cream or buttermilk then season with salt and pepper to taste and serve.
If serving the soup cold, remove from heat, cool, stir in the cream or buttermilk, and refrigerate covered.
Season to taste with salt and pepper.
Serve hot or cold.