Prepare the Gnocchi Dough: In a large bowl, whisk together the pumpkin puree, beaten egg, grated Parmesan, salt, nutmeg, and pepper until smooth. Gradually add 2 cups of flour, stirring with a fork until a shaggy dough begins to form.

Knead the Dough: Turn the dough out onto a well-floured surface. Gently knead, adding the remaining 1/2 cup of flour a little at a time only as needed, until the dough is soft, slightly tacky, but no longer sticky. Avoid over-kneading. Shape the dough into a ball.

Shape the Gnocchi: Divide the dough into 4 equal parts. Roll one piece into a long rope, about 3/4-inch thick. Use a sharp knife or bench scraper to cut the rope into 1-inch pieces. You can leave them as little pillows or roll them over the tines of a fork to create ridges. Place the shaped gnocchi on a floured baking sheet. Repeat with the remaining dough.

Cook the Gnocchi: Bring a large pot of generously salted water to a boil. Working in batches, gently drop the gnocchi into the boiling water. They are done when they float to the surface, about 2-3 minutes. Let them cook for another 30 seconds once they float. Use a slotted spoon or spider strainer to remove them directly to the pan with the finished sauce.

Make the Sauce: While the water is boiling, make the sauce. In a large skillet or pan over medium heat, melt the butter. Add the sage leaves and cook for 1-2 minutes, until fragrant and crispy. Remove the sage leaves and set aside for garnish. Continue swirling the butter until it foams and then turns a golden amber color with brown bits at the bottom, emitting a nutty aroma. Immediately remove from heat.

Combine and Serve: Once the cooked gnocchi are transferred to the brown butter sauce, gently toss to coat. If the sauce is too thick, add a tablespoon of the starchy pasta water to loosen it. Season with additional salt and pepper if needed.

Garnish and Serve: Divide among plates, garnish with the crispy sage leaves, extra Parmesan, and black pepper. Serve immediately.