If your mashed potatoes are quite wet, start by spreading them out in a frying pan over medium low heat to dry them out. This may take up to 10 minutes depending on the type of potato and amount of liquid used. If you used russet potatoes and didn't add much liquid you can skip this step.Pour the flour into a large, shallow bowl and push towards the outside. Then add the potatoes and also push outwards so that there is a well in the middle. (If your potatoes just came off the stove, let them cool slightly before proceeding to the next step.)Next, add the beaten egg and salt into the potato well. Using a fork, start gently incorporating the potatoes inwards, followed by the flour. When a loose dough starts to form that isn't too sticky, start using your hands to grab half of the dough and fold it over itself; push down gently to compact the dough a little and then give the bowl a quarter turn before repeating. Some of the flour from the bowl won't be folded in yet - that's OK! If the dough sticks to your fingers, pick up some of the loose flour and sprinkle it over the dough before each fold. If it just feels wet, but not terribly sticky, leave the flour at the bottom of the bowl.After about 15 folds, cut the dough in half. On a well-floured counter top roll half of the dough into a log approximately 3/4 inch in diameter. Using a knife or pastry scraper cut the log into 1/2 inch sections. Toss the pieces around in some flour so that the exposed sticky ends don't glue themselves to the counter. Lay the pieces out on a baking tray until ready to cook. Repeat with the other half of the dough.To cook, bring a large pot of heavily salted water to a boil. At the same time get a large frying pan and brown the butter over medium heat. When the butter is just barely starting to change colour, add the majority of the tarragon and turn down to low until ready to use so that the butter doesn't burn.Work in batches. Take 1/3 of the gnocchi and add to the boiling water. Turn the butter up to medium-high. After only a couple of minutes the dumplings will float to the top; remove with a slotted spoon, allowing most of the water to drip off, and transfer immediately to the butter pan. Brown for about 3 minutes per side; the outside will be crisp. Try to time it so that you're constantly adding fresh dumpling to the butter pan so that everything can be served hot.Serve with a few pieces of fresh tarragon, grated parmesan, and freshly ground pepper.