Split the lobster and section it. Remove the claws and break them so that the meat can be easily removed when the lobster is cooked. Keep the liver and the coral. Put the pieces of lobster in a deep frying pan with 4 tablespoons of hot oil and just over 1 oz. of butter. Cook till the shell has turned red, and the meat of the lobster is quite firm. Strain off the fat, sprinkle with the finely chopped shallots, add the brandy, the white wine, the fish stock, the demi-glace sauce, the tomato purée, a little chopped parsley, and a dash of cayenne. Cover the pan and put in a moderate oven for 15 to 20 minutes. Remove the meat from the lobster shell and keep hot while the sauce is being prepared. Reduce the sauce by simmering to ½ a pint. Add the chopped lobster coral, a small piece of butter, cook for 2 or 3 minutes, and strain. Replace the sauce on the fire without letting it boil, and when hot remove from the fire and stir in a ¼ lb. of butter, divided in small pieces. Pour the sauce over the pieces of lobster and sprinkle the dish with a little chopped parsley.