Add the flour, pumpkin puree, egg and salt to a large bowl. Mix until a shaggy dough forms. Turn onto a lightly floured surface and knead until smooth and elastic, about 8 to 10 minutes. If the dough is sticky, work in additional flour a little at a time.
Cover the dough and let it rest at room temperature for 30 minutes.
Place the dough on a lightly floured surface and divide into 6 equal pieces. Keep the pieces covered while you work so they don't dry out.
Flatten one piece slightly and run it through your pasta machine on the widest setting. Fold it and run it through again 2 to 3 times. Continue passing the dough through the machine, moving to a thinner setting each time, until it reaches your desired thickness. Repeat with the remaining dough.
Lightly flour each pasta sheet and run it through the fettuccine cutter. Toss the noodles lightly with flour to prevent sticking.
Bring a large pot of generously salted water to a boil. Add the fresh pasta and cook for about 2 to 3 minutes, or until tender. Reserve about ½ cup of the pasta water, then drain.
While the pasta cooks, melt the butter in a large skillet over medium heat. Add the sage and cook until fragrant and beginning to crisp. Add the garlic and cook for about 30 seconds.
Reduce the heat to low. Add the heavy cream and Parmesan and stir until the cheese has melted and the sauce is smooth. Add a splash of reserved pasta water as needed to loosen the sauce.
Add the cooked pumpkin pasta to the skillet and gently toss until thoroughly coated. Taste and add salt if needed. Serve immediately with additional Parmesan and crispy sage, if desired.