Preheat barbeque or grill on high heat
To prepare the lobster, place the live lobster into a freezer for an hour, this will put the lobster to sleep.
Once the lobster is completely chilled remove from the freezer and using a sharp knife stab the lobster through the head and cut in half lengthwise through the shell, place the lobster onto a tray flesh side up.
For the butter, place the room temperature salted butter into a medium size mixing bowl, finely chop the tarragon leaves and place into the bowl, add in the seeded mustard and lemon zest season with salt and pepper to taste and mix well until all ingredients are well combined, remove the butter from the bowl and place into cling film or baking paper, roll into a log and using the edges of the cling film tighten the log to remove the air, place the butter into the fridge or freezer until set.
Season the lobster halves with salt and pepper, cut the tarragon butter into 1 cm medallions and place 2 medallions on each half lobster.
Place the lobsters onto the preheated barbeque or grill flesh side up and close the lid to keep the heat in, cook for 8-10mins or until the flesh is just translucent in the centre and the all the butter has melted and started to caramelise, remove from grill and serve with lemon wedges.