Fill a stock pot with water and heat over high. Season liberally with salt and bring to a boil while you prep other ingredients.

Heat 2 Tbsp. oil in a Dutch oven or large high-sided skillet over medium heat. Arrange mushrooms in a single layer and cook, undisturbed, until bottom side is golden brown, about 3 minutes. Toss and continue cooking 2 to 3 more minutes, until softened. Transfer to a bowl.

Add remaining 2 Tbsp. oil and onion to pan. Cook, stirring often, until onion is translucent, about 5 minutes. Add garlic, salt, red pepper flakes, and nutmeg; cook 1 minute, until fragrant.

Add tomato paste and stir to evenly coat onion. Continue to cook, stirring often, until paste is deep red and starting to brown on the bottom of the pan, about 5 minutes. Add vodka to deglaze pan, stirring to scrape up browned bits of tomato paste. Cook 1 minute to allow some of the alcohol to cook off.

Stir in pumpkin puree, coconut milk, and sautéed mushrooms; reduce heat to low.

Once pasta water is boiling, add pasta and cook until almost al dente. Add kale during final minute of cooking.

Use a slotted spoon to transfer cooked pasta and kale to skillet with vodka sauce. Stir in pasta cooking water in 1/2 cup increments until sauce turns smooth and glossy and coats each piece of pasta. Stir in fresh thyme leaves and garnish with Parmesan, if desired.