Scrub the potatoes well and cut them into halves. Place in a pot of cold water and bring to a boil.
Once the potatoes are tender, remove them from the water and quickly slip off their skins. Mash thoroughly while still hot and spread the mash out slightly so it cools.
Once the potato mash is still slightly warm but not hot, make a well in the center and add the egg. Sprinkle with 3/4 of the flour and use a bench scraper to scrape the potato edges into the eggy middle. If the dough is very sticky gradually add more flour until it is a cohesive, delicate dough.
Divide the dough into fourths and roll each piece into a log about 1/2" thick. Cut into 3/4" pieces and roll each piece on the tines of a fork to get the classic gnocchi shape.
Bring some salted water to a boil. Drop the gnocchi in about ten at a time. When they float to the surface, count to ten, then remove them with a large slotted spoon or spider to a serving plate. Once all the gnocchi is cooked, pour on a sauce of your choice (I used a fresh homemade tomato sauce) and top with basil chiffonade and freshly grated Parmesan cheese.