Meanwhile, remove the meat from your cooked lobster, keeping the shell to one side, and slice the meat into small, bite sized chunks. Place the chopped meat into a bowl for later.
Salt the now boiling water, then add the peeled and chopped potatoes, along with the leftover lobster shell pieces. Boil the potatoes for 15 minutes until soft.
Meanwhile, heat a frying pan to a low temperature. Add 1 tablespoon of butter to the pan, then crush the garlic clove lightly with the back of a knife and add that to the pan. Add the chopped lobster meat to the pan to warm for 4 to 5 minutes, moving the lobster around the pan regularly, then remove from the heat.
Drain the pot containing the potatoes, discard the lobster shell, and return the cooked potatoes to the pot.
Using a potato masher, mash the potatoes until they are smooth. Add the remaining 2 tablespoons of butter, the cream, and the tarragon to the pan, add salt and pepper to taste, and mix everything until combined.
Gently transfer the warmed lobster to the potatoes, along with any leftover juice in the frying pan, and fold them in.
Top the lobster mashed potatoes with fresh chives and serve.