Step 1

Preheat oven to 425 degrees F (220 degrees C).

Step 2

Drizzle 1 tablespoon olive oil in the bottom of a large roasting pan. Place beef, pork, and chicken in pan and turn to coat with olive oil.

Step 3

Roast in the preheated oven until meat is well browned, 20 to 30 minutes.

Step 4

Heat remaining 1 tablespoon olive oil in a large pot over medium heat. Cook and stir onion with a pinch of salt in hot oil until onion is soft and translucent, about 5 minutes. Add garlic; cook and stir until fragrant, about 1 minute.

Step 5

Pour crushed tomatoes, 1 1/2 cups water, and tomato paste into onion mixture. Add roasted beef, pork, and chicken to tomato sauce mixture.

Step 6

Pour remaining 1/2 cup water into the roasting pan, and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Pour roasting pan water mixture into tomato mixture. Stir Parmigiano-Reggiano cheese, Italian parsley, 2 teaspoons salt, ground black pepper, and red pepper flakes into tomato sauce; bring to a simmer, reduce heat to low, and simmer gently until sauce reduces and meat is tender, about 4 hours. Transfer meat to a dish. Adjust sauce seasonings to taste.