In a large pot, melt the butter over medium-high heat. Add the onions, celery, and cayenne and cook, stirring, until soft, about 4 minutes. Add the garlic and cook for 30 seconds. Add the mushrooms, thyme, salt, and pepper, and cook, stirring, the mushrooms give off their liquid and start to brown, about 7 minutes. Add the brandy, bring to a boil and cook until glazed, about 2 minutes. Add the stock and return to a boil. Reduce the heat to medium-low and simmer uncovered, stirring occasionally, for 15 minutes.
Remove from the heat and puree with a hand-held immersion blender, or in batches in a food processor. Add the cream, return to a simmer, and cook for 5 minutes. Remove from the heat and adjust seasoning, to taste.