Preheat oven to 425°F (220°C). Toss the cubed butternut squash with olive oil, salt, and pepper on a rimmed baking sheet. Spread in a single layer and roast for 20-25 minutes, flipping halfway, until tender and caramelized on the edges.

While squash roasts, place the diced pancetta in a large cold skillet. Set over medium heat and cook, stirring occasionally, until the fat renders and the pancetta is crisp and reaches an internal temperature of 145°F (63°C), about 6-8 minutes. Use a slotted spoon to transfer pancetta to a paper towel-lined plate, leaving the rendered fat in the skillet.

Add the butter to the same skillet with the rendered fat. Cook over medium heat, swirling often, until the butter foams, turns golden, and gives off a nutty aroma, 3-4 minutes. Immediately drop in the sage leaves and sliced garlic. Fry for 30-60 seconds until the sage is crisp and garlic is golden, then remove from heat. Transfer the crisp sage leaves to a plate with the pancetta.

Bring a large pot of generously salted water to a boil. Cook the casarecce according to package directions until al dente. Reserve 1 cup of pasta cooking water, then drain.

Return the skillet with the brown butter (garlic still inside) to low heat. Add the drained pasta, roasted squash, and half the crispy pancetta. Toss well, adding splashes of reserved pasta water to create a glossy, creamy sauce that clings to the noodles. Remove from heat, fold in the grated Parmigiano-Reggiano, and season with salt, pepper, and red pepper flakes if using. Serve immediately, topped with remaining pancetta, the crisp sage leaves, extra Parmesan, and a drizzle of olive oil.