Thinly slice shallots and smash garlic cloves. Put all the rest of the ingredients in a pan and bring to the boil. Remove from the heat and leave them to cool.
Cut all the vegetables for the court bouillon into 1 inch pieces, put all the rest of the ingredients into a saucepan and bring to the boil.
Prepare the lobsters by removing the tail and the claws from the head. Keep any coral, place to one side. Remove the intestine from the lobster tail. Cook the tail in the court bouillon for 1 minute and the claws for 1 minute just to release the meat from the shell. Remove the meat from the shell and chill.
Take the lobster heads and cut into 2 inch pieces, cover with diced butter and bake in the oven at 150C for 1.5 hours. When done carefully strain the oil off, leave any residue in the bottom of the tray and discard the shells.
Preparing the vegetables; turn the baby artichokes placing in acidulated water to keep their colour. Cut into 4 and remove the choke, drain off well. Pour some olive oil in a hot pan, add the artichokes, thyme and garlic season with salt. Cover with a lid and shake to release all the flavours. Add the white wine, cover again and cook until the wine is reduced. Then add the chicken stock. Cover with a lid and continue to cook on a high heat until they are ready. There should be very little liquid left in the pan. Place on a tray and leave to cool.
Then prepare the rest of the vegetables, keep to one side 1 radish, 1 navet, 1 carrots (these are to be sliced raw for a garnish). Blanch the rest in boiling salted water individually until each one is perfectly cooked then refresh in ice water. Drain well. Add pickling liquor to separate containers and place vegetables in the separate containers so the colours do not mix. Then chill.
Blend the lobster coral and water together then place in a sous vide bag and poach at 95C until it is bright pink and cooked. Cool. Pass the coral through a medium sieve. The mixture should resemble lobster eggs. Finely chop the shallots and combine with the rest of the ingredients. Adjust salt and lemon to taste.
Prepare a vinaigrette by mixing the ingredients together.
Prepare tomato concasse, salad leaves and herbs, thinly slice raw vegetables. Slice lardo di colonnata.
Heat the lobster butter to 56C, poach the tails and the claws for 10-15minutes..
Before serving each lobster dress with vinaigrette and season. Strain the vegetables. Dress the plates.
Add diced tarragon and tomato to the coral dressing at the last second.