Preheat the oven to 400°F.

Pierce the potatoes all over with a fork. Bake until very tender, about 1 hour. Remove and let cool.

When the potatoes are cool enough to handle, halve them lengthwise and scoop out the flesh, discarding the skins. Press the potato flesh through a ricer or food mill into a large bowl (or mash well). Add the flour and 1 ½ teaspoons salt and stir to combine. Stir in the egg yolks and mix until a dough comes together.

Line a baking sheet with parchment paper and dust the parchment with flour. On a lightly floured surface, turn out the dough and divide it in half. Using your hands, shape each half into a long rope about ¾ inch thick. Slice the ropes crosswise into ¾ inch-long pieces. If desired, roll each piece over the tines of a fork to create ridges. Place the gnocchi on the prepared baking sheet. Let dry for about 20 minutes.

Bring a large pot of salted water to a boil. Add half of the gnocchi and cook until they rise to the surface, about 4 minutes. Using a slotted spoon, transfer the gnocchi to a plate. Repeat with the remaining gnocchi.

Heat a large skillet over medium-low heat. Add the butter and cook, tilting the pan constantly to move the butter around, until melted. Continue to cook as the butter foams and starts to brown, swirling the pan occasionally, 3 to 4 minutes more. Add the sage and cook until fragrant and crispy, about 1 minute.

Add the boiled gnocchi to the butter sauce and stir gently to coat and warm through, 1–2 minute. Season with salt and pepper, then divide among bowls. Top with the grated Parmesan and serve.