Heat 1/4 cup extra-virgin olive oil in a saucepan. Add the butternut squash, salt, and pepper and saute until the butternut squash is fork tender. Transfer the squash mixture to a bowl. Using the back of a fork, mash until the squash is an even mixture. Let cool.
Add the ricotta cheese, 1 cup Parmigiano cheese, flour, and an egg. Mix thoroughly until the mixture becomes a soft dough. Cook's Note: You may need to add more flour if the mixture is too moist.
Using your hands, scoop enough of the mixture to roll into the size of a golf ball. Repeat with the remaining mixture.
Place the gnudi balls in a pot of boiling salted water for about 1 minute, or until they float to the surface. Drain and plate.
Drizzle the gnudi balls with the remaining extra-virgin olive oil and sprinkle with Parmigiano cheese.