Bring a large pot of heavily salted water to a rolling boil. In a medium saucepan, combine the cubed butternut squash, diced shallot, smashed garlic cloves, and 2 cups of vegetable broth over medium-high heat. Bring to a boil, then reduce heat to medium-low, cover, and simmer for 12 to 14 minutes until the squash is fork-tender.

While the squash is simmering, add the rigatoni to the boiling salted water and cook until 1 to 2 minutes shy of al dente (about 10 to 11 minutes). Before draining, dip a measuring cup into the pot to reserve 1 cup of cloudy, starchy pasta water. Drain pasta and set aside.

Carefully transfer the cooked butternut squash, garlic, shallot, and simmering broth into a high-speed blender. Add the heavy cream, ground nutmeg, 3/4 teaspoon salt, and black pepper. Blend on high speed for 60 to 90 seconds until completely silky, glossy, and lump-free.

In a large, deep skillet or Dutch oven, melt the 4 tablespoons of butter over medium heat. Once foaming, add the whole fresh sage leaves in a single layer. Fry for 2 to 3 minutes, swirling the pan, until the sage is crisp and the butter turns golden amber with a fragrant, nutty aroma. Immediately transfer the crispy sage leaves to a paper towel-lined plate, leaving the nutty browned butter in the pan.

Pour the velvety blended butternut squash puree directly into the skillet with the brown butter over medium heat. Stir to emulsify the butter into the puree until rich and glossy.

Add the drained al dente rigatoni into the sauce. Pour in 1/4 cup of the reserved hot pasta water. Toss and stir vigorously for 2 minutes over medium heat, allowing the sauce to cling tightly to the outer ridges and penetrate the hollow pasta tubes.

Turn off the heat. Fold in the freshly grated Parmigiano-Reggiano, stirring continuously until melted into the sauce. If the pasta seems too thick, splash in an extra tablespoon of pasta water to achieve a glossy restaurant sheen.

Divide warm rigatoni among shallow pasta bowls. Garnish each serving with crispy fried sage leaves, toasted chopped walnuts, shaved parmesan ribbons, and freshly cracked black pepper. Serve immediately.