Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.

Toss the cubed butternut squash with olive oil, 1/2 tsp salt, and 1/4 tsp black pepper. Spread it in a single layer on the prepared baking sheet.

Roast for 30-35 minutes, or until the squash is very tender and slightly caramelized. Remove from oven and let cool slightly.

Mash the roasted squash using a potato ricer or fork until completely smooth. You want a very dry puree. If it seems wet, spread it back on the baking sheet and bake for another 10-15 minutes to dry it out further. Measure out 1.5 cups of puree and let it cool completely to room temperature.

In a large mixing bowl, combine the cooled butternut squash puree, egg yolk, 1 tsp salt, and grated nutmeg. Mix well.

Gradually add 2.5 cups of the all-purpose flour, mixing with a wooden spoon or your hands until a shaggy dough forms. Turn the dough out onto a lightly floured surface.

Knead the dough very gently for 2-3 minutes, adding more flour (up to 1/2 cup) as needed, until it just comes together into a soft, slightly sticky but manageable dough. Avoid overworking, as this will make the gnocchi tough. The dough should be soft, but not sticking to your hands or the counter.

Divide the dough into 6-8 equal portions. Roll each portion into a long, thin rope, about 1/2 inch in diameter. If the dough is too sticky, dust your hands and the surface with a little flour.

Cut each rope into 3/4-inch pieces. For traditional gnocchi texture, roll each piece lightly over a gnocchi board or the tines of a fork to create ridges, or leave them smooth. Place the finished gnocchi on a lightly floured baking sheet, making sure they don't touch.

Bring a large pot of heavily salted water to a rolling boil.

Working in batches, carefully drop the gnocchi into the boiling water. Do not overcrowd the pot. The gnocchi will sink, then float to the surface when cooked. Once they float, let them cook for an additional 1-2 minutes to ensure they are cooked through. This usually takes 3-4 minutes total per batch.

Use a slotted spoon to remove the cooked gnocchi from the water, allowing excess water to drain. Transfer them directly to the skillet with the brown butter sauce, or to a plate while you prepare the sauce.

While the gnocchi are cooking (or immediately after), melt the unsalted butter in a large skillet or sauté pan over medium heat.

Continue cooking, stirring occasionally, until the butter foams, turns golden brown, and smells nutty (about 4-6 minutes). Be careful not to burn it.

Add the fresh sage leaves to the brown butter and cook for 1-2 minutes until crispy. Stir in 1/2 tsp salt and 1/4 tsp black pepper.

Add the cooked gnocchi directly to the skillet with the brown butter and sage. Gently toss to coat the gnocchi evenly.

Serve immediately, garnished with freshly grated Parmesan cheese.