Put the potatoes in a 6-quart / 5.7 L pot, cover with cold water, and boil whole with the skin on for 35 to 40 minutes until a knife slides in with no resistance.

Drain and let the potatoes cool just until you can handle them, then peel while still warm and pass through a potato ricer onto a clean work surface. Working with warm potatoes keeps the dough dry.

Spread the riced potato out, sift the chestnut flour and 00 flour over the top, sprinkle with the salt, and pour the beaten egg over the mound.

Work the ingredients together with a bench scraper or your hands just until a soft, slightly tacky dough forms, about 1 minute. Chestnut flour has no gluten to toughen, so stop as soon as the dough holds together, over-kneading makes it gummy.

Cut the dough into 4 pieces. Roll each piece on a lightly floured surface into a rope about 2 cm / 3/4 inch thick.

Cut each rope into 2 cm / 3/4 inch pieces. Roll each piece over the tines of a fork or a gnocchi board for ridges, or leave them plain.

Bring a 6-quart / 5.7 L pot of generously salted water to a boil. Drop the gnocchi in batches of about 20 so the water stays at a boil, stirring once gently so they don't stick to the bottom.

Cook until the gnocchi float to the surface, 2 to 3 minutes, then let them bob for another 20 to 30 seconds before lifting out with a slotted spoon. Reserve a splash of the cooking water.

While the first batch cooks, melt the butter in a 12-inch / 30 cm skillet over medium heat. Add the sage leaves and cook for 3 to 4 minutes, swirling the pan, until the butter turns golden brown and smells nutty.

Transfer the drained gnocchi straight into the skillet with the browned butter. Add a splash of the reserved cooking water and toss gently to coat.

Scatter in the toasted walnuts and half the Parmigiano, tossing for 1 minute over low heat so the cheese just melts into the butter.

Plate immediately, finish with the remaining Parmigiano and a few grinds of black pepper.