Spread the pumpkin puree onto a double layer of paper towels. Press another layer on top and squeeze gently to remove as much moisture as possible until the puree is thick and concentrated.
In a large mixing bowl, combine the drained pumpkin puree, lightly beaten egg, grated Parmesan cheese, nutmeg, and salt. Stir until well blended.
Gradually add the all-purpose flour to the pumpkin mixture, 1/2 cup at a time. Use your hands to gently fold the flour in until a soft, slightly tacky dough forms. Be careful not to over-knead, as this will make the gnocchi rubbery.
On a floured surface, divide the dough into four equal portions. Roll each portion into a long rope about 3/4-inch thick. Use a knife to cut the ropes into 1/2-inch long pieces. Optionally, roll each piece over the tines of a fork to create ridges.
Bring a large pot of salted water to a gentle boil. Working in batches, drop the gnocchi into the water. Cook until they rise to the surface, which takes about 1 to 2 minutes. Use a slotted spoon to remove them and set aside.
While the water boils, heat a large skillet over medium-high heat. Add the crumbled chicken sausage and cook until browned and cooked through, about 5-7 minutes. Remove the sausage from the skillet and set aside.
In the same skillet, melt the butter over medium heat. Once it begins to foam, add the sage leaves and garlic. Cook for 2-3 minutes until the butter turns a golden nutty brown and the sage leaves become crisp.
Stir in the chicken broth and black pepper to deglaze the pan, scraping up any browned bits. Add the cooked gnocchi and chicken sausage back into the skillet.
Toss gently for 1-2 minutes until the gnocchi are lightly seared and coated in the sage butter sauce. Serve immediately with extra Parmesan cheese if desired.