Bring a large pot of generously salted water to a rolling boil. Cook the fettuccine according to the package directions until just al dente. Reserve 120 ml of starchy pasta water before draining.

While the pasta cooks, melt the butter in a wide skillet over medium heat. Add the sage leaves and swirl frequently as the butter foams and crackles. Continue cooking until the milk solids turn a golden amber brown and smell nutty, about 3 to 4 minutes. Remove the crispy sage leaves with a fork and set aside for garnish.

Reduce heat to medium-low, add the minced garlic to the browned butter, and sauté for 30 seconds until fragrant. Whisk in the pumpkin puree, vegetable broth, heavy cream, nutmeg, salt, and black pepper until completely smooth. Let the sauce simmer gently for 2 to 3 minutes.

Transfer the drained fettuccine directly into the skillet with the pumpkin sauce. Add the grated Parmigiano-Reggiano and a splash of reserved pasta water. Toss vigorously with tongs over low heat for 1 to 2 minutes until the sauce turns glossy and clings tightly to the pasta.

Divide the pasta among warm shallow bowls. Crumble the reserved crispy sage leaves over top, finish with freshly cracked black pepper, and serve immediately with extra grated cheese.