Bring a large pot of salted water to boil and add pasta according to package directions. Cook until al dente, approximately 9-10 minutes, then drain and set aside, reserving 1 cup pasta water for sauce adjustments.
While pasta cooks, heat olive oil and butter in a large skillet over medium heat. Add diced onion and cook for 3 minutes until softened, then add minced garlic and sauté for another minute until fragrant.
Pour in vegetable broth and pumpkin puree, stirring constantly to combine smoothly. Add dried sage, thyme, nutmeg, and red pepper flakes, cooking for 3-4 minutes until the sauce begins simmering gently.
Reduce heat to low and slowly stir in heavy cream, mixing until the sauce reaches silky consistency. The creamy hearty pumpkin feta pasta sauce should look glossy and luxurious at this stage.
Gently fold in crumbled feta cheese, stirring carefully to distribute throughout without breaking up the pieces too much. Taste and adjust seasonings with salt and pepper as needed, keeping the feta's natural saltiness in mind.
Add cooked pasta to the sauce and toss gently to coat each piece evenly. If the sauce seems too thick, add reserved pasta water one tablespoon at a time until reaching your preferred consistency.
Transfer to serving bowls and top with grated Parmesan cheese and extra red pepper flakes. Serve immediately while steam still rises from the dish.