Warm 8 tablespoons of oil in a frying pan and fry the chopped shallots; add some white wine and let it evaporate. Cut the claws of the lobster and mash lightly with the help of a meat mallet, cut the lobsters in half lengthwise. Put in the lobster claws in the pan with the sauce and cook for 2 minutes. Remove the lobster from the pan and the claws and extract the pulp, taking care not to break the shell. Cut the flesh of the lobster and of the claws into cubes. Blanch the tomatoes for 10 seconds in boiling water, drain them with the help of a slotted spoon and cool them in iced water. Then spell them and cut them into cubes. Put in the pan the diced lobster and tomatoes and cook for 5 minutes, then add salt and pepper. Blanch the parsley leaves in boiling water for 1 minute, drain and cool in ice water, drain again and put the parsley, the remaining oil and a little ice and blend everything with a blender, add salt and pepper . Cook the potato gems in salted water, drain and put them in the pan with the sauce by adding 4 tablespoons of the parsley pesto. Serve the potatoes in the shell of the lobster and garnish with parsley pesto.