Cook pasta in a large pot of boiling, salted water until al dente. (I generally use 4 qts of water with 2 Tbs kosher salt.) Drain well, reserving 1 1/2 cups of the water. Sprinkle with olive oil and toss.
Over medium low heat, mix pumpkin puree, garlic, milk, apple cider, parmesan, vinegar, pepper, salt, basil, nutmeg, and cinnamon, and 3/4 to 1 cup of pasta water to pot. Stir sauce until heated through-5 to 10 minutes.
Check sauce seasoning and add additional salt to taste.
Add pasta and sauce to pasta pot and toss to coat. If sauce is too thick, add some additional pasta water.