Begin by bringing a large pot of salted water to a boil.

While waiting for the water to boil, prepare the potatoes by placing them in a saucepan and covering them with cold water. Bring to a boil over high heat, then reduce the heat to medium-low and simmer for 15-20 minutes, or until they're tender when pierced with a fork.

Drain the potatoes and allow them to cool slightly. Then, use a potato masher or ricer to mash them into a smooth consistency.

In a separate mixing bowl, combine the mashed potatoes, flour, and salt. Mix well.

Add the egg to the mixture and stir until fully incorporated.

Gradually add the melted butter, stirring constantly until the dough comes together.

Turn the dough out onto a floured surface and knead for several minutes until it becomes smooth and elastic.

Divide the dough into four equal parts and roll each piece into a long rope shape.

Cut each rope into 1-inch pieces and press the tines of a fork into each one to create the characteristic gnocchi texture.

Remove the gnocchi from the water with a slotted spoon or tongs and transfer them to a plate. Serve immediately, topped with your favorite sauce and a sprinkling of Parmesan cheese.