Heat oil in a large pot over medium-high. Add mushrooms and cook undisturbed for 3-4 minutes. Stir and let fry another 3-4 minutes. Repeat until mushrooms have browned nicely (~10 minutes total).
Lower heat to medium then add onion and celery and cook until onion has softened and turned translucent (~3-5 min).
Stir in garlic and carrots until fragrant (~30 sec) then pour in sherry and cook down a couple more minutes.
Add tapioca and 1 cup of broth and stir until starch is a dissolved slurry. Pour in the rest of the chicken broth then bring to a boil. Reduce heat to medium low and simmer, stirring frequently until the soup is thickened to your liking (~20 min.)
Stir in wild rice, thyme, rosemary, and red pepper flakes. Return to simmer.
Stir in coconut cream and cook until warmed through, but do not let it boil. Lower heat if necessary (~20 minutes).
Season with salt and pepper to taste.