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Bring a large pot of salted water to a boil. Meanwhile, heat olive oil in a Dutch oven or large high-sided skillet over medium heat. Add mushrooms and onion; cook until softened, about 8 minutes. Season with ½ tsp. salt.
Add tomato paste, garlic, red pepper flakes, and nutmeg; mix to combine. Cook until the tomato paste caramelizes and turns brick red, about 3 minutes.Add vodka to deglaze the pan, stirring to scrape up browned bits. Cook 3 to 5 minutes to allow some of the alcohol to evaporate. Stir in pumpkin puree, coconut milk (or heavy cream), remaining ½ tsp. salt, and a few grinds of black pepper; reduce heat to low.
Once pasta water is boiling, add rigatoni and cook until almost al dente. Add kale during final minute of cooking. (Note: if you're using spinach, add it to the vodka sauce rather than the boiling pasta water. Spinach needs much less heat and cooking time.)
Use a slotted spoon to transfer cooked pasta and kale to the pan with vodka sauce. (Or, reserve 1 cup pasta cooking water, and drain pasta and kale.)Stir in pasta cooking water in ½ cup increments until the sauce turns smooth and glossy and coats pasta (I use 1 cup total). Stir in fresh thyme leaves and garnish with Parmesan, if desired.