Preheat the oven to 375 degrees F.

Rub the cut slices of the squash with 2 tablespoons of the olive oil and season with 1/2 teaspoon of the salt and 1/4 teaspoon of the white pepper. Set the squash, cut side down, on a baking sheet and roast until tender, 45 minutes to 1 hour. Remove from the oven. When cool enough to handle, scoop the flesh from the squash and place in a large bowl, discarding the skin. Mash into a chunky puree with a potato masher, then let cool, about 15 minutes.

Add the remaining 1/4 cup of olive oil, 1/2 teaspoon of the salt, remaining 1/4 teaspoon white pepper, oregano, chopped sage and nutmeg, and work into the warm squash. Sprinkle the flour over the mixture and using your hands, work into a smooth, soft dough. Cover the dough with a damp kitchen towel and let rest for 30 to 45 minutes.

Turn onto a lightly floured surface and divide into 4 equal portions. Roll each portion between your palms into a long rope, about 3/4-inch thick. If the rope won't hold together, return it to the bowl with the remaining dough and work in more flour, 1 teaspoon at a time, as needed. Cut each rope of dough into 3/4-inch long pieces. Press each piece against the tines of a floured fork, then drop into the lightly floured surface. (Gnocchi also can be placed in a single layer on a lightly floured baking sheet and refrigerated for several hours. Alternately, the gnocchi can be frozen for 1 hour then transferred to an airtight container and frozen for up to 1 month.)

In a large pot, bring 4 quarts of water and the remaining 2 teaspoons of salt to a low boil.

Add the gnocchi in batches and cook just until they float to the top, 1 1/2 to 2 minutes (or 3 minutes for frozen gnocchi.)

For the Balsamic Brown Butter: In a medium saucepan, cook the butter over medium heat until brown bits form and it has a nutty aroma, about 3 minutes. Remove from the heat and add the shallots, balsamic vinegar, and sage. Stir well. Adjust seasoning, to taste.

Remove the gnocchi from the water with a slotted spoon and dry on towels. Transfer to a large serving dish and gently toss to coat with the balsamic butter. Garnish with black pepper, parsley, chives, and shaved Parmesan