Preheat oven to 425°F. Prick potatoes all over with a fork; place on a rimmed baking sheet lined with parchment paper. Roast in preheated oven 15 minutes.

Meanwhile, trim top and bottom from unpeeled squash; cut squash in half crosswise where neck meets base. Place both halves upright on cutting board on widest flat cut side; carefully cut in half from top to bottom. Scoop out and discard seeds. Rub squash all over with 2 teaspoons oil; sprinkle evenly with pepper and ¼ teaspoon salt.

Arrange squash, cut sides down, around potatoes on baking sheet. Return to oven, and roast at 425°F until squash and potatoes are easily pierced with a knife, about 45 minutes. Remove from oven; let cool slightly, about 10 minutes.

Cut potatoes in half lengthwise. Scoop out potato flesh, and pass through a ricer into a medium bowl. (Alternatively, mash potato flesh using a potato masher in a medium bowl until smooth.) Spread potatoes on a large plate lined with paper towels. Pat dry with paper towels. Set aside 1 ½ cups packed mashed potatoes. Reserve remaining potatoes for another use.

Scoop flesh from cooled squash into a medium bowl; mash until smooth. Spread squash across a large plate lined with paper towels. Pat dry with paper towels to remove as much excess moisture as possible to yield 1 ¼ cups mashed squash.

Whisk together chestnut flour, 00 flour, and 1 teaspoon salt in a large bowl; create a large well in center of mixture. Add egg yolks, half of potatoes, half of squash, half of grated cheese, and a pinch of nutmeg to well. Repeat layering with remaining potatoes, squash, cheese, and nutmeg.

Gently fold flour mixture into potato mixture until a shaggy mass forms. Transfer dough to a work surface. Knead briefly until evenly incorporated.

Line two large baking sheets with parchment paper, and dust parchment with 00 flour. Place a small bowl of 00 flour near workstation. Cut dough evenly into 8 wedges. Dust wedges with 00 flour. Gently roll 1 dough wedge into a ¾-inch-thick rope. Cut rope crosswise into ¾-inch-long gnocchi. Transfer gnocchi to prepared baking sheets. Repeat with remaining dough wedges.

Bring a large pot of water to a boil over high. While water comes to a boil, heat butter in a large skillet over medium just until butter starts to foam. Add sage and remaining ¼ teaspoon salt; cook, stirring often, until foam subsides and sage is sizzling, about 3 minutes. Pour mixture into a small bowl. Set aside until ready to use.

Salt boiling water, and gently stir in half of gnocchi. Cook, undisturbed, until gnocchi float to surface, 2 to 3 minutes.

While gnocchi cook, heat 2 tablespoons oil in a large nonstick skillet over medium-high. Using a slotted spoon or a spider, transfer gnocchi to skillet. Cook, undisturbed, until gnocchi bottoms are crisp, about 4 minutes. Flip gnocchi, and cook until other side is crisp, about 2 minutes. Transfer gnocchi to a plate. Do not wipe skillet clean. Repeat with remaining 2 tablespoons oil and remaining gnocchi.

Return all cooked gnocchi to skillet; pour butter mixture over gnocchi, and stir to coat. Season with additional salt to taste. Divide gnocchi, butter sauce, and fried sage evenly among plates. Garnish with additional cheese.