How to Make It
Preheat the oven to 350°. Pierce the potatoes all over with a knife, set them on the bottom rack of the oven and bake for about 1 hour, or until tender. Meanwhile, set the butternut squash, cut side down, on a lightly oiled baking pan. Cover with foil and bake on the upper rack of the oven for about 50 minutes, or until tender. Let the squash cool slightly, then peel and cut into 1-inch dice.
Split the potatoes, scoop out the flesh and pass it through a ricer or coarse sieve into a large bowl. Stir in the egg yolk and 1/2 teaspoon salt. Add the flour, 1/4 cup at a time, stirring to form a crumbly dough. Turn the dough out onto a lightly floured work surface and knead until smooth. Wrap in plastic and let rest at room temperature for 30 minutes.
Line a large rimmed baking sheet with wax paper and dust the paper lightly with flour. Divide the dough into quarters. Working with 1 piece at a time and keeping the rest loosely wrapped in plastic, roll each piece into a 1/2 -inch-thick rope. Using a lightly floured knife, cut the rope into 1/2 -inch lengths. Spread the gnocchi out on the wax paper and toss to coat with flour. Roll and cut the remaining gnocchi dough and let stand uncovered at room temperature.
In a large skillet, melt 1 tablespoon of the butter in the oil. Add the mushrooms, season with salt and pepper and cook over high heat until tender, about 7 minutes. Transfer to a plate. Add the remaining 5 tablespoons of butter to the skillet and cook the butter over high heat until golden, 2 to 3 minutes. Add the lemon juice and sage and cook for 30 seconds longer. Add the squash and cook, stirring, just until coated with the butter. Add the mushrooms, season with salt and pepper and keep warm.
Meanwhile, bring a large pot of salted water to a boil. Add the gnocchi and cook, stirring once or twice, until they rise to the surface and the water returns to a boil, about 1 minute. Drain the gnocchi and add them to the squash. Stir to coat with the butter. Transfer to a large, deep platter and serve immediately.