Place potatoes in a large, deep pot and cover with cold water. Add a pinch of salt and boil for 30 minutes or until potatoes are tender.
Dust a clean work surface with extra flour. Use a potato ricer to rice potatoes, discarding skins, in an even layer onto floured surface. Pour egg over potato, then sift half of the flour over egg. Use a butter knife in a chopping motion to roughly combine mixture. Use hands to gently knead dough, using remaining flour (and extra flour, if required), until it comes together and is no longer sticky.
Clean work surface and lightly dust with a little more extra flour. Cut dough into 8 equal pieces, then roll each into a 2cm-thick log. Cut each log into 2cm pieces using a sharp knife.
Heat a large, deep frying pan over medium heat. Add butter and cook until foaming. Add sage and cook, stirring occasionally, for 2 minutes or until sage is crispy and butter has started to darken in colour. Remove pan from heat.
Bring a large pot of water to the boil over medium-low heat, and season well with salt. Add gnocchi and boil for 30 seconds or until gnocchi rises to the surface.
Place sage mixture over medium heat. Using a slotted spoon, add gnocchi to pan and cook for 4 minutes, gently tossing occasionally, or until gnocchi is golden brown. Season and serve with parmesan.
If you don't have a potato ricer, allow potatoes to cool slightly, then remove and discard skins, and grate potatoes instead.