Slice the peeled and boiled potatoes in half and put them in a potato ricer. Rice all of the potatoes. If using a grater, grate all of the potatoes into the bowl. I recommend either of these methods over mashing, because it will not overwork the starch of the potato. If you don't have a grater or a ricer, you can mash the potatoes with a fork or masher, but make sure not to mash them too much or they will become gluey!
In a large bowl, lightly combine the potatoes with your hands. Add the salt and mix into the potato mixture. Then, create a well in the middle of the mashed potato mixture.
Crack an egg into the well and with a fork, mix the mashed potato mixture and the egg together until the egg is fully combined in the mashed potatoes.
Place the mashed potato mixture onto a clean and floured work surface and flatten.
Pour a quarter of the flour onto the potato mixture and knead the flour into the potato mixture. Keep flouring your workspace so the dough doesn't stick when you knead it.
Repeat the kneading with quarter sections of the flour, until the dough feels tender, fluffy and as though it's absorbed enough flour. You don't want it to be too firm or too sticky. Use your instinct!
When the dough is done, roll it into a large ball. With a bench scraper or knife, cut the large ball into quarters.
On your floured workspace, with your hands, roll each quarter of dough out into a long log that is about 3/4 inch thick (give or take, make it whatever thickness you like.)
Cut the log into 1-inch(ish) fluffy pillows. Repeat with the next quarter of dough until all of the dough is cut into gnocchi. Set all of the gnocchi aside.
You can boil the gnocchi this way, or you can jazz up the shape by rolling each piece onto a fork to round it out and give it divots. The choice is yours! If refrigerating, place the gnocchi on a sheet pan and cover with plastic wrap or tin foil.