Preheat the oven to 375 degrees F. Pour a mound of kosher salt onto a rimmed baking sheet. Prick the potatoes all over with a fork and nestle them in the salt; this prevents the potatoes from browning and draws out their moisture. Bake the potatoes until fork-tender, about 45 minutes. Let sit until just cool enough to handle.

Halve the potatoes lengthwise and scoop the flesh into a potato ricer; discard the skin. Press the flesh through the ricer onto a floured counter or cutting board; let cool 3 to 4 minutes.

Spread out the potatoes slightly. Beat the egg with 1/2 teaspoon salt in a small bowl and drizzle over the potatoes.

Sprinkle the potato mixture with 1 cup flour and knead until a smooth dough forms, adding up to 1/2 cup more flour if the dough is sticky. Cover with a kitchen towel and let rest 10 minutes.

Lightly flour your surface. Divide the dough into 6 pieces and roll into "snakes," about 1/2 inch in diameter. Cut into 1/2-to-1-inch lengths with a bench cutter or knife; transfer to a parchment-lined baking sheet and lightly dust with flour.

Make the sauce: Melt the butter in a large skillet over high heat. Cook, stirring, until the butter develops brown flecks and smells nutty. Watch it closely to keep the butter solids from burning. Remove from the heat.

Bring a large pot of salted water to a boil. Add the gnocchi; wait until they float to the surface, then simmer 30 more seconds. (Taste one to make sure it's cooked through.) Return the brown butter to medium-low heat. Scoop the gnocchi from the pot with a spider or slotted spoon, transfer to the skillet and toss until some of the brown-butter flecks stick to the gnocchi. Add the red pepper flakes, sage and a few tablespoons of the gnocchi cooking water; toss again. Remove from the heat and add half of the cheese. Divide among bowls and top with the remaining cheese.

Photograph by David Malosh