Notes: Make the soup at least 3 1/2 hours before dinner, or up to 1 day ahead. Marinate the mushrooms at least an hour, or up to 1 day ahead; bring to room temperature to serve. Mix the trout cakes up to 4 hours ahead (cover and chill), but shape and fry them at the last minute. PREP AND COOK TIME: About 1 1/4 hours, plus at least 2 hours to chill soup.
Pour olive oil into a 3- to 4-quart pan over medium-high heat. When hot, add onion and stir often until limp, 4 to 5 minutes. Add asparagus and 2 1/2 cups broth. Bring to a boil over high heat, then reduce heat and simmer, stirring occasionally, until asparagus is very tender when pierced, 10 to 14 minutes.
Working in batches and holding lid down with a towel, whirl asparagus mixture in a blender until very smooth. Pour into a large bowl and stir in cream. Add salt and pepper to taste. Cover and chill until cold, at least 2 hours.
If soup is too thick, stir in a little more broth. Taste, and add more salt and pepper if desired. Ladle soup into six wide, shallow bowls. Set a warm trout cake in the center of each, and top cakes equally with marinated morels. Drizzle a little truffle oil over soup around cakes. Serve immediately.