Detach the head from the tail, remove the claws and joints. Chop the head and set aside (3/4 for the fumet and 1/4 for the clarification). Insert a bamboo skewer into the tail. Cook in a steam oven at 100°C for 8 minutes. Shell the tail whilst hot, remove the digestive tract, trim off any filaments, and roll tightly in cling film to obtain a straight cylindrical shape. Refresh in iced water then refrigerate. Leave the large claw for a further minute. Once cooked, refresh the claws in iced water, shell, remove the cartilage, cut the meat into a small 3 mm salpicon and set aside.
Remove the cling film from the lobster tail. Cut into 5 mm thick medallions. Chill in a blast freezer then spray with jellied fumet to give shine. Repeat again if necessary then refrigerate. For lobster salpicon, just before assembling the dish, coat the lobster with vinaigrette and season with salt and Espelette pepper. Add the finely chopped tarragon at the very last minute. Refrigerate.
Peel the avocados, remove the stone then blend the flesh with the lime juice, and a dash of ascorbic acid, until smooth. Correct the seasoning with salt, transfer to a piping bag and refrigerate.
Cut the aromatic garnish (carrot, onion, garlic) into a mirepoix and sweat in olive oil without colouring. Cook for 5-6 minutes then add the fresh tomato and tomato purée, sweat and cook to remove the tomatoes’ acidity. Add the chopped lobster head, sweat then add the bouquet garni and fish fumet to cover. Gently simmer for 30 – 40 minutes until the fumet takes on a rich flavour. Strain through a china cap sieve and skim off any surface fat, if necessary. Chill in a blast freezer.
Combine the clarification ingredients with the crushed lobster head. Add the cold lobster fumet and heat to 60°C. Gently simmer without stirring. Make three holes in the surface to enable the fumet to be skimmed whilst cooking. When the fumet is crystal-clear and transparent, correct the seasoning with salt and Espelette pepper and add the gelatine that has been softened in cold water. Filter through a muslin cloth. Set aside at room temperature.
Over a bowl of ice, whip the cream to stiff peaks, season with salt. Transfer to a piping bag fitted with a star shaped nozzle. Refrigerate.
Spoon a 1.5 cm layer of seasoned lobster salpicon into the bottom of a small plastic glass (5.5 cm high x 2.5 cm wide). Use a cork, which has been covered in kitchen foil, to smooth and even out the surface. Chill in a blast freezer then pour a 1.5 cm layer of avocado cream on top. Gently tap the glass to smooth the surface, return to the blast freezer then, using a piston funnel, add a 1 cm (or 15 g) layer of lobster jelly on top. Chill in a blast freezer to set then position a lobster medallion on top with a dot of whipped cream in the centre. Decorate with lumpfish roe, a tarragon sprig and sprinkle over the Espelette pepper.