Heat butter and oil in a medium pot over medium-high heat.
Sauté onion, garlic, and mushrooms for ~6-7 mins, or until tender, stirring occasionally. Reserve some of the mushrooms for topping the soup, if desired.
Add thyme, flour, salt and pepper and cook for an additional 1-2 mins, stirring frequently, until fragrant.
Add cream and cook for a minute or so, stirring frequently.
Add broth and stir in uncooked rice. Bring to a boil, then reduce heat to medium-low, cover and simmer until rice is tender (~35 mins).
Adjust salt and pepper, then serve immediately, topping each bowl with sautéed mushrooms and cilantro, as preferred.
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