Peel the potatoes, cut into 1" cubes, and boil in 4 litres/1gallon of water until fork-tender (10-15 minutes)
Drain in a colander and let rest a few minutes to let the steam out. We want fairly dry potatoes.
Using a ricer, or a potato masher and a lot of elbow grease, process the potatoes until completely smooth.
Stir in the salt, and butter. Taste for saltiness and adjust.
Sprinkle the corn starch and work it into the potatoes to make a dough.
If the mixture is too crumbly, I would recommend adding a beaten egg.
See below for what to do next because the possibilities are near-infinite.