1. 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.
2. 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.
3. 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.