Bring a large pot of water to boil on the stove and set up a large ice bath near the pot.
Place the mashed potatoes on a clean surface and make a well in the center of the potatoes. Distribute the flour evenly over the potatoes.
Crack the egg into the well, and beat with a fork. The egg will slowly start to pull the flour and potato mixture into the center. Continue beating until the egg is no longer separate from the flour and potato mixture.
Roll entire mixture into a ball and knead for 4 - 5 minutes until dry. Use additional flour sparingly if mixture remains too moist.
Separate dough into 4 equal portions and roll each portion into a long rope about 3/4" in diameter. Cut each rope into 1" segments and pinch each segment slightly for texture.
Once all gnocchi are formed, place in batches into the boiling water. Cook until gnocchi float to the surface, about 1 minute, then remove with a slotted spoon and place in the ice bath. Continue until all gnocchi are cooked and placed in ice bath.
In a large saute pan, melt butter over medium heat. Allow to sit until milk solids separate to bottom of pan and begin to turn brown. Immediately turn off heat and add sage and gnocchi. Toss until gnocchi are thoroughly coated with butter and sage mixture. Serve immediately.