Step 1

Preheat oven to 350°F. Place potatoes directly on middle rack in preheated oven. Bake until very tender when pierced with a paring knife, 1 hour and 15 minutes to 1 hour and 30 minutes. Carefully cut potatoes in half lengthwise; let stand until cool enough to handle, about 20 minutes. Using a paring knife, peel potatoes, and discard skins.

Step 2

Press potatoes through a potato ricer (or thoroughly mash in a bowl and then place) onto a clean work surface; spread in an even layer. Let cool until almost room temperature, about 8 minutes. Drizzle egg yolk evenly over potatoes. Sprinkle salt and 1/2 cup of the flour evenly over potatoes and egg yolk. Using a bench scraper or a rubber spatula, very gently fold mixture over onto itself several times, gradually folding in remaining 1/2 cup flour just until all flour is incorporated and dough comes together in a mass. Do not knead or overwork.

Step 3

Line a baking sheet with parchment paper, and lightly dust with flour; set aside. Clean work surface, and lightly dust with flour. Divide dough evenly into 4 pieces, about 5 1/4 ounces each. Roll 1 dough piece into a 3/4-inch-thick rope, about 9 inches long. Cut rope crosswise into 3/4-inch-long gnocchi pieces. Roll each piece against tines of a fork to make ridges; transfer to prepared baking sheet. Repeat process with remaining 3 dough pieces.

Step 4

Bring salted water to a boil in a 4-quart stockpot over high. Add half of gnocchi to boiling water, and gently stir once. Cook until gnocchi rise to surface, about 2 minutes, and then gently boil 1 more minute, reducing heat as necessary to maintain a gentle boil. While gnocchi cooks, melt 3 tablespoons of the butter in a large skillet over medium-high. Using a slotted spoon, transfer cooked gnocchi to skillet; add 3 tablespoons gnocchi cooking liquid. Cook, stirring constantly, until a creamy and loose butter sauce forms and coats gnocchi, about 1 minute. Remove from heat, and season with salt and pepper to taste; place in 2 serving bowls.

Step 5

Repeat Step 3 with remaining gnocchi and 3 tablespoons butter. Sprinkle each serving with grated cheese and nutmeg. Serve immediately.