Pulse the flour and salt together in a food processor. Add in the egg, egg yolks, pumpkin puree, and sage. Pulse again until the mixture comes together into a ball.
Place the dough on a lightly floured surface and divide it into four equal portions. Roll each portion out into a thin rectangle shape (no need to be precise) of approximately ¼" in thickness. *Note - Cover the other portions of dough with plastic wrap, so they don't dry out until you're ready to roll them.
Use a pizza cutter or pasta wheel to cut the dough into strips, approximately all of the same thickness. Lay the pasta strips out onto a parchment-lined baking sheet, making sure they have space in between so they don't stick to each other. Proceed with the same rolling and cutting process with the remaining dough pieces.
If cooking immediately:
If cooking immediately, bring a large pot of water to boil along with approximately 2 tablespoons of kosher salt. Drop in the pasta and cook for approximately 2 minutes or until it floats to the water's surface. Drain. Stir in the butter and top with Parmesan cheese. Serve immediately.
If drying the pasta for later:
If drying, cover the pasta strips loosely with paper towels and allow the pasta to air dry for at least 8-10 hours or overnight. When ready to cook, bring a large pot of water to boil with approximately 2 tablespoons of kosher salt. Drop the dried pasta into the boiling water and cook for 10-12 minutes or until it floats to the water's surface. Drain. Stir in the butter and top with Parmesan cheese. Serve immediately.
Notes
This pasta may be cooked immediately after making, or it may be dried for later use.Store the dried pasta in a cool, dry place in an airtight container for up to 5 days.