Prep all ingredients: dice onion, mince garlic, grate Pecorino, and chop sage. Set aside for now. Place a large pot of water on the stove to boil.
Begin the sauce while the water is coming up to boil. Add olive oil and butter to a large skillet set over medium-low heat. Once the butter is melted and everything is well heated, add the diced onion and cook for 5 minutes, stirring occasionally.
Add the minced garlic to the skillet and cook for 1 minute. Season with paprika, kosher salt, and black pepper and stir in.
Sprinkle flour over the onion and garlic and stir in. Cook for 2 minutes, stirring often so it doesn't stick to the bottom of the pan, then slowly whisk in the broth and milk. Stir and cook as the sauce thickens until it can coat the back of a wooden spoon.
Once the pot of water is boiling, salt it and add your pasta. Cook according to package directions, reserving a ladle or two of water at the end of cook time before draining it.
When the sauce has thickened, turn off the heat and scatter in the grated Pecorino and whisk as it melts to combine. Whisk in the pumpkin puree, chopped sage, and cayenne (if using), then add the drained pasta and toss. Add some of the reserved pasta water to thin the sauce to your liking, if desired.
Taste for salt and pepper. Serve with extra grated Pecorino, sage, and/or a grating of fresh nutmeg.