Preheat the broiler. On a rimmed baking sheet, toss the squash, oil and pumpkin spice. Season with salt and pepper and arrange in a single layer on the baking sheet. Broil, turning once, until the squash is tender and golden brown, about 8 minutes.
Meanwhile, bring a large pot of water to a boil; salt the water, add the pasta and cook until al dente. Drain the pasta, reserving 1 cup of the starchy cooking water.
In a large skillet, melt the butter over medium heat. Add the sage leaves, rosemary leaves and thyme leaves; cook until the leaves are crispy and the butter is browned, about 5 minutes. Using a slotted spoon, transfer the sage to paper towels; season. (Leave the rosemary and thyme in the butter.)
Toss the pasta with the brown-butter mixture in the skillet. Stir in the chopped herbs, cream and the grated Parmesan, adding the reserved pasta water, 1/4 cup at a time, as needed to make a thick sauce. Add the squash, season and gently toss. Transfer the pasta to bowls; top with the fried sage leaves and shaved Parmesan.
Rachael Ray Every Day's October 2016 cover