While crawfish is cooking, melt remaining 3 tablespoons of butter in a medium saucepan. Stir in mushrooms, remaining 3 tablespoons garlic, salt, and pepper. Cook until mushrooms have released their moisture, about 5-7 minutes. Add flour (or arrowroot) while continuously stirring and cook for about 30 seconds. Stir in stock and cream and simmer until sauce is thick and reduced, about 12 minutes.
Add creamed mushrooms to crawfish. Add paprika and cayenne pepper and stir. Season to taste with salt and pepper.