To make the meatballs, place the meat in a large bowl, add the Parmigiano, egg yolk, breadcrumbs, garlic and parsley, and season well with salt and pepper. Mix thoroughly but delicately so as not to compress the meat, which would toughen it.

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 sauté until translucent , about 4 minutes. Add the mushrooms, season with salt and pepper, and sauté 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.