Take a piece of the meat mixture about the size of a small grape and roll the mixture between the palms of your hands into a small ball. Place on a plate, and repeat with the remaining meat mixture.
Line a large shallow plate with paper towels. Set aside.
Heat 1 cup of the olive oil in a 10- to 12-inch skillet over medium heat. Place only as many meatballs in the pan as you can without crowding them and panfry until browned and crisp on one side, about 3 minutes. Flip and cook until the other side is browned and crisp, about 2 minutes more. Remove the meatballs from the pan with a spatula and place on the prepared plate to drain. Repeat with any remaining meatballs.
Heat the remaining 2 tablespoons olive oil in another large skillet over medium-high heat. Add the onion and saut until translucent , about 4 minutes. Add the mushrooms, season with salt and pepper, and saut 5 minutes. Add the peas, toss, and cook until the mushrooms release their liquid, about 2 to 3 minutes. Add the stock and mint and cook for another 5 minutes. Taste and, if necessary, adjust the seasoning. Add the meatballs, toss well, and set aside, covered, to keep warm.
Fill a 10-quart stockpot with 7 quarts of water. Add 2 tablespoons salt and bring to a boil. Add the pasta and cook until al dente.
Reserve 1/2 cup of the cooking liquid, then drain the pasta and stir it into the sauce. Add the butter, the Parmigiano, and a few tablespoons of the reserved cooking water, and heat over very low heat. Taste and adjust the seasoning, then toss thoroughly, drizzling with more olive oil if desired.