Add unpeeled, whole potatoes to a large pot. Add cold water and fill pot so that water fills about 1 inch above potatoes. Place on stove and set to medium-high so that water begins to boil. Boil until potatoes are completely cooked, about 45 minutes. While potatoes boil, you can work on making the pesto (see instructions below).
Once cooked, remove potatoes from water (but do not toss water; you will use it to boil gnocchi) and immediately peel and pass through a potato ricer. Place riced potato on a clean counter surface.
Sprinkle flour all over potato. Make a well in the center of the potato and flour mixture. Place egg in center of well. Using a fork, scramble egg and incorporate into potato and flour mixture. Once egg is mixed it, use your hands to incorporate the rest of the potato and flour mixture until a smooth ball forms.
Cut dough in 8 sections. Grabbing one section at a time, roll into a long log, about 2 feet long. Cut log into 1 inch pieces. Roll each piece into an oval to form the dumpling. Repeat with 7 other sections. Add a little flour to surface if the dough gets too sticky.
Boil gnocchi until it floats to the top (about 1 minute). Do not over-crowd the pot with gnocchi. Only place enough dumplings to make one layer of gnocchi. See below on steps on how to serve the gnocchi.