Preheat oven to 350°F. Place potatoes directly on middle rack in preheated oven. Bake until very tender when pierced with a paring knife, 75-90 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. Press potatoes through a potato ricer (or thoroughly mash in a bowl).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 ½ 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 ½ cup flour just until all flour is incorporated and dough comes together in a mass. Do not knead or overwork.

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 ¼ ounces each.

Roll 1 dough piece into a ¾-inch-thick rope, about 9 inches long. Cut rope crosswise into ¾-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.

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.