Split the lobster tails in two down the middle. Melt the butter in a saucepan and pour in the oil. Fry the lobster for about 2 minutes on each side. (If you cook it too long, it becomes too chewy, so watch the pan carefully.).
Mix the gremolata ingredients, season with salt and pepper and mix in the lemon juice.
Divide the tomatoes among 4 plates. Put the lobster on top and add the sauce from the pan.
Note: The key to cooking a whole lobster is that the tails and claws have different cooking times. It is recommended to cut the lobster into pieces before cooking it. You could ask your fish dealer to do it for you at the market.