Bring a large pot of water to a boil and salt it generously. Cook pasta according to package directions until al dente. Before draining, reserve 1 cup of the starchy pasta water.
While the pasta cooks, prepare the brown butter sage sauce. In a large skillet over medium heat, melt the butter. Once melted, continue cooking, swirling occasionally, until the butter turns amber in color and smells nutty, about 3-4 minutes.
Add the sage leaves to the browned butter and cook for 30 seconds until they become crisp. Remove about half of the sage leaves and set aside on a paper towel for garnish.
Reduce heat to medium-low and add minced garlic to the brown butter. Cook for 30 seconds until fragrant, being careful not to burn it.
Stir in the pumpkin purée and whisk until smooth. Cook for 1-2 minutes to remove any raw pumpkin flavor.
Pour in the heavy cream and whisk until fully incorporated with the pumpkin mixture. Allow to simmer gently for 2-3 minutes until slightly thickened.
Season the sauce with nutmeg, cinnamon (if using), salt, pepper, and red pepper flakes if desired. Taste and adjust seasonings as needed.
Add the drained pasta directly to the sauce, along with 1/4 cup of the reserved pasta water. Toss to coat the pasta evenly, adding more pasta water if needed to achieve your desired consistency.
Sprinkle in the Parmesan cheese and toss until melted and incorporated into the sauce.
Serve immediately, topped with the reserved crispy sage leaves, additional Parmesan cheese, and freshly ground black pepper.