Sweat the vegetables (carrot, onion, celery, shallot, leek, and fennel) in butter without browning them. Add the crushed tomatoes and tomato purée, leave to cook and add the bouquet garni. Sweat. Add the crushed lobster bodies, sweat well and cover with water. Add the rice. When the fish stock tastes right, drain, strain and reduce if necessary. Cream, bring to the boil and make any adjustments to the seasoning with salt, pepper and Espelette pepper. Stir in the gelatine that has been previously softened, cool a small quantity of bisque to check the seasoning when cold.
Remove the tomato seeds without blanching, blend and dilute very slightly using the tomato juice. Season with salt and white pepper, and add the tabasco. Beat the tomato coulis with the olive oil. Pass through a fine strainer and gelatinize with 16 g of gelatine per litre.
Whisk nine-tenths of the cream into a Chantilly, and gently fold in the mascarpone. Infuse the rest of the warmed cream with tarragon for 20 min, stir in the softened gelatine, strain. Leave to cool a little, then fold this mixture into the Chantilly. Adjust the seasoning of salt and pepper, transfer to a piping bag fitted with a n°8 star tip nozzle.
Roll the puff pastry out to 1.5 mm thick. Using a spatula, spread the tapenade over one half of the pastry, and fold the other half over it. Roll the pastry out again to 2 mm thick, cool in a blast chiller, and cut into batons 3 cm long and 3 mm thick. Freeze for a short time, gently twist, and place on a sheet of baking parchment. Bake at 180°C (356°F) until golden brown and flaky.
Pour the room-temperature bisque into a glass using a 3 cm (20 g) piston funnel, leave to gelatinize in the fridge. Pour 9 mm of tomato coulis over the top, set using a blast chiller, and a dash of tarragon chantilly in the middle of the glass. Top with a lobster medallion (previously glazed) and a dab of lumpfish roe. Just before serving, garnish the top with a sprig of dill and add a little feuilleté on the side.