Fill a large pot with water, salt it generously, and bring it to a rolling boil. Add the pasta and cook until just al dente, stirring occasionally. Reserve about a cup of pasta water before draining, since it helps loosen the sauce later.
Melt the butter in a skillet over medium heat and let it foam, then continue cooking until it turns golden and smells nutty. Add the whole sage leaves and let them crisp for about a minute. Remove the leaves with a slotted spoon and set them aside for garnish.
Lower the heat and stir the chopped sage, garlic, and chili flakes into the browned butter. Cook for about a minute until fragrant, watching closely so the garlic doesn't burn. Stir in the pumpkin puree and apple cider right after.
Let the pumpkin and cider mixture simmer gently for a few minutes so the flavors concentrate. This step matters more than it looks, since it cooks off the raw pumpkin taste. Keep the heat low and stir every so often.
Pour in the milk, then whisk in the parmesan and gouda a handful at a time. Season with nutmeg, salt, and pepper, tasting as you go. The sauce is ready once it looks glossy and coats the back of a spoon.
Add the drained pasta straight into the skillet and toss until every piece is coated. Loosen with a splash of reserved pasta water if the sauce feels too thick. Finish with the crispy sage leaves and extra parmesan on top.