Preheat the oven to 375 degrees F. Using a vegetable peeler, peel the butternut squash into thin ribbons; don’t worry if they vary in size.
Spread the squash ribbons in a single layer on a parchment-lined baking sheet, season with a pinch of salt and drizzle with the olive oil, then roast for about 10 minutes until just beginning to soften.
Meanwhile, start cooking the fettuccine according to the box directions.
Preheat a cast-iron or non-stick skillet over medium heat and add the butter.
When the butter melts, add the mushrooms, kale, and garlic; season with a pinch of salt and the dried oregano and sauté for 2 minutes until the vegetables soften slightly.
Add the heavy cream and reduce the heat to medium-low, then season the cream mixture with salt and black pepper.
Once the pasta is done, drain it and reserve about 1/2 cup of the pasta water to adjust the sauce consistency.
When the roasted squash is ready, add the ribbons to the cream and vegetable mixture.
Toss the cooked pasta into the creamy vegetable mixture and thin the sauce with some of the reserved pasta water until it reaches your preferred thickness.
Stir in the grated Parmesan and a pinch of red pepper flakes if using, then toss to combine.