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.