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

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.

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

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.

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

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.