Blanch the crayfish in boiling water for 3-4 minutes, drain and let cool slightly. Remove the tail meat and devein.
Rinse the soup vegetables, trim and chop. Peel the onion and cut into quarters. Arrange the vegetables in a roasting pan along with the crayfish shells. Drizzle with oil and roast in a preheated oven at 250°C (approximately 475°F) for 15 minutes. Add the stock and roast for another 15-25 minutes. Strain through a fine-mesh sieve.
Melt the crab butter in a large Dutch oven. Peel the shallot and chop finely. Sauté in the butter along with the caraway and flour. Deglaze with the wine then pour in the fish stock and simmer for 3-5 minutes, stirring. Season with salt and pepper then stir in the sugar and Cognac. Add the crayfish tails and gently warm.