Make the Gnocchi: Boil the potatoes with their skins on in a pot of salted water until they are tender all the way through. Once cooked, peel the potatoes while they are still hot and pass them through a potato ricer or mash them until smooth. On a clean surface, make a mound with the potato, creating a well in the center. Add the beaten egg and a sprinkle of salt to the well. Gradually incorporate the flour into the potato and egg mixture, kneading gently until a soft dough forms. Be careful not to over-knead. Divide the dough into sections and roll each into long ropes, about 3/4 inch in diameter. Cut the ropes into 1-inch pieces and press each piece with a fork or gnocchi board to create ridges. Bring a large pot of salted water to a boil. Cook the gnocchi in batches until they float to the surface, then remove them with a slotted spoon.

Prepare the Sauce: In a large skillet, melt the butter over medium heat. Add the sage leaves and cook until the butter starts to brown and the sage becomes crispy. Add the cooked gnocchi to the skillet, tossing gently to coat with the sage butter. Season with salt and pepper to taste.

Serve: Plate the gnocchi and drizzle any remaining butter and sage over the top. Sprinkle with freshly grated Parmesan cheese before serving.