Remove the crawfish from the shell. Cut a slit along the back of the crawfish to devein them. Rinse the crawfish thoroughly, then pat dry.
Peel the garlic and crush into a paste with a little salt. Mix the garlic paste with the olive oil. Add to the crawfish and stir until combined.
Bring the fish stock and the Noilly Prat to a boil. Mix in the double cream, reduce the heat, and simmer until reduced by half.
Remove the crawfish from the marinade, then pat dry with paper towels. Cook the crawfish on an electric grill pan (or in a regular grill pan) quickly on both sides over high heat.
Mix the caviar into the sauce, then stir to combine. Serve the crawfish with the sauce.
The crawfish fit well with sautéed vegetables as a side.