Rinse the crawfish in cold water. Peel them, reserving all the shells. Set aside about 1/2 cup of tail meat. Chop the rest of the tails finely in a food processor.
Crush the crawfish shells with a meat mallet. Put the crawfish shells, garlic cloves, thyme and bay leaf into a quart of cold water in a saucepan, and bring to a simmer for 30 minutes. Strain, and reduce the stock down to about one cup.
Heat the olive oil in a large skillet over medium heat, and sauté the onions, carrot, and parsley until soft. Add the tomato puree and cook another minute or so. Then add the wine, bring to a boil, and reduce until most of the liquid is gone.
Add the chopped crawfish tails to the pan. Add the brandy, bring to a boil, and carefully flame it (if you like). Add one cup of the crawfish stock to the pan and bring to a light boil.
Make a blond roux with the butter and flour. Whisk into the sauce until completely blended. Add the cream, salt, and cayenne, adjusting the latter two to taste. Bring to a simmer, then serve. Garnish with the reserved whole tails.
