1 pound chicken livers
6 tablespoons extra-virgin olive oil
3 tablespoons butter
1 medium onion, finely chopped
2 bay leaves
1 teaspoon kosher salt, plus more for the pasta water
3 cups chicken stock, hot
1 tablespoon Tomato sauce
1 cup frozen peas
freshly ground black pepper, to taste
⅓ cup Italian parsley, freshly chopped
1½ cups Parmigiano-Reggiano, freshly grated
Clean the chicken livers, removing all the fat, veins, and membranes. Chop them into small bits with a chef's knife.
Heat 3 tablespoons of the olive oil and all the butter in the large skillet over medium heat. Stir in the onion and the bay leaves, and season with 1/4 teaspoon salt. When the onion is wilted and translucent, stir in the chopped chicken livers, and salt again lightly.
Pour in 1/4 cup hot stock, raise the heat, and cook, stirring, until the moisture is nearly evaporated. Clear a space in the skillet, drop in the tomato paste, toast it in the hot spot for a minute, then stir the paste in with the chopped livers and onion. Scatter the frozen peas into the skillet, stir, then pour in the remaining stock.
Bring the liquid to a boil, and adjust the heat to keep it bubbling steadily. Season with 1/2 teaspoon salt and generous grinds of black pepper. Let the sauce cook and reduce, stirring occasionally, until most of the moisture is gone and it has a thick, flowing consistency, 10 minutes or so. Adjust the seasonings, and have the sauce simmering when the pasta is dressed.
Meanwhile, fill the large pot with 8 quarts of water, add 2 tablespoons salt, cover, and heat to a rolling boil. Shake excess flour off the bigoli, drop into the pot, and return rapidly to the boil. Cook, occasionally stirring, until the pasta is nearly al dente (time will vary with the thickness of the strands). Lift the bigoli from the water with a spider, drain briefly, and turn it into the simmering sauce. Toss together until the pasta is fully cooked and dressed with sauce.
Turn off the heat, toss in the parsley, the remaining 3 tablespoons olive oil, and a cup of the grated cheese. Serve immediately, passing more cheese at the table.