Heat 1 tablespoon olive oil and 1 tablespoon butter in a large deep skillet over medium heat. Add cubed butternut squash, minced garlic, and 1/2 teaspoon kosher salt. Sauté for 6 to 8 minutes until edges turn lightly caramelized. Pour in 1 cup vegetable broth, bring to a simmer, cover with a lid, and cook for 10 minutes until fork-tender.

Meanwhile, bring a large pot of generously salted water to a rolling boil. Cook the rigatoni pasta until just shy of al dente (about 1 minute less than package instructions). Crucially, scoop out and reserve 1 full cup of starchy pasta water before draining.

Transfer the tender simmered butternut squash and cooking broth to a high-speed blender (or use an immersion blender directly in the skillet). Add heavy cream and freshly grated nutmeg. Blend on high speed for 60 seconds until silky-smooth and velvety golden-orange.

Wipe out the skillet and place over medium heat. Add the remaining 2 tablespoons of butter. Once foaming, add fresh sage leaves in a single layer. Fry for 90 seconds until crackling-crisp and the butter turns a fragrant nutty hazelnut-brown. Immediately transfer crispy sage leaves to a paper towel, leaving the infused brown butter in the skillet.

Pour the velvety butternut puree into the brown butter in the skillet over low heat. Add the cooked rigatoni and freshly grated Parmigiano-Reggiano. Toss vigorously with tongs, splashing in 1/4 to 1/2 cup of reserved pasta water until the glossy sauce clings tightly to every ridge. Season with black pepper and salt. Serve in shallow bowls topped with crispy sage, shaved parmesan, and toasted pecans.