Peel, cut in small cubes and boil the potatoes in lightly salted water until they become really soft (about 20 minutes).
Mash the potatoes evenly and add the finely chopped tarragon leaves and the flour when they are still warm (overwise they'll become sticky). Mix well until you get a nice non-sticky dough.
When you have an even dough similar in consistency to the bread dough, cut it in smaller parts and roll them with your palms on a wooden board to get a long snake thick about 2 cm (less than 1 inch) of diameter. (Keep the surface and the dough smooth with flour to prevent stickiness).
Now with a dough-cutter, cut the gnocchi about 2-3 cm long (approx 1 inch). (You can then roll them on a fork or on a gnocchi board to get the typical gnocchi texture). Let them rest on a clean towel sprinkled with flour.
In a large pot, bring to boil salted water and then toss the gnocchi one by one so they won't stick to each other. It will take 1-2 minutes to cook. Meanwhile, melt the butter in a pan.
As soon as the gnocchi float to the surface, it means they are ready. Take them out with a skimmer and toss them in the pan with the melted butter.
Plate the gnocchi and on top add the grated ricotta salata and ground black pepper to garnish.