Roast the squash and make the gnocchi a day ahead and then this dish may be put together quickly. I often use purchased potato gnocchi from the chill cabinet in my local supermarket and it is still delicious.
Plate the grated, cooked potatoes on a work surface in a mound. Make a well in the center and sprinkle with salt and pepper. Add the egg yolk and begin to knead in half the flour. When moistened, add the other half of flour and knead until the dough comes together. let it rest for 2 minutes.
Fill a large pot with cold water, add 1 teaspoons salt and bring to a boil. Have a bowl of ice water nearby.
Cut the dough into 4 equal pieces. Roll each piece into a 1/2 inch log. Cut each log into 1/2 inch pieces. When the water is boiling drop the gnocchi into the water and boil until the pieces float to the surface, about 1 minute.
Remove the gnocchi and plung into the ice water to stop the cooking, then let drain.
Melt butter in a 10 inch skillet over medium high heat. Stir in mushrooms and cook until tender, about 5 minutes. Stir in cubed roasted squash and cooked potato gnocchi. Add lemon juice, sage strips and season with salt and pepper.
Cook, tossing occasionally until gnocchi and squash start to brown, about 5 to 8 minutes.