Boil the water. Bring 4 quarts of water to a boil in a large pot. Add 1 tablespoon salt.

Start the sauce. While the water heats, melt the butter with the olive oil in a 12-inch deep skillet over medium heat. Add the onion and cook 4 to 5 minutes, stirring, until soft and translucent but not browned.

Add garlic and sage. Stir in the garlic and sage and cook 1 minute, until fragrant.

Cook the pumpkin. Add the pumpkin puree and cook 2 minutes, stirring often. This removes some of its raw, canned taste and deepens the color.

Add the liquid. Whisk in the broth. Bring to a gentle simmer and cook 5 minutes, stirring occasionally, until slightly thickened.

Cook the pasta. While the sauce simmers, cook the pasta in the boiling water 1 minute short of the package time (it will finish in the sauce). Scoop out 1 cup of pasta water, then drain.

Finish the sauce. Reduce the heat to low. Stir in the cream, nutmeg, 1 teaspoon salt, the black pepper, and red pepper flakes if using. Add the Parmesan a handful at a time, stirring until each addition melts. Do not let the sauce boil after adding the cheese.

Toss. Add the drained pasta and 1/2 cup of the pasta water. Toss over low heat for 1 to 2 minutes, adding more pasta water a splash at a time until the sauce coats the pasta and looks glossy.

Brighten and serve. Stir in the lemon juice, taste, and adjust the salt and pepper. Serve right away with extra Parmesan and parsley.