In a large bowl, mix together the pumpkin puree and egg until smooth. Add the flour, salt, and nutmeg, stirring until a soft dough forms. Don't overmix—just bring it together!
Turn the dough out onto a floured surface and divide it into 4 portions. Roll each portion into a long rope about 3/4-inch thick.
Cut each rope into 1-inch pieces and gently press each piece with a fork to create ridges (this helps the sauce stick!). Place finished gnocchi on a floured baking sheet.
Bring a large pot of salted water to a boil. Working in batches, drop the gnocchi into the boiling water. When they float to the top (about 2-3 minutes), they're done! Remove with a slotted spoon.
While the gnocchi cooks, melt the butter in a large skillet over medium heat. Add the sage leaves and cook until the butter turns golden brown and smells nutty, about 4-5 minutes.
Add the cooked gnocchi to the brown butter and toss gently to coat. Season with salt and pepper, then serve immediately with plenty of Parmesan cheese on top!