Pat the pork cubes dry with paper towels and season with 1 tsp of the salt and the pepper. Dice the onion, carrots, and celery to about 5 mm so they melt into the broth.

Heat the olive oil in a 28 cm / 11-inch Dutch oven over medium-high heat. Brown the pork in two batches, 5 to 6 minutes per batch, turning until most sides are deep brown. Move each batch to a bowl.

Reduce the heat to medium and add the onion, carrots, celery, and a pinch of salt. Cook for 8 minutes, scraping up the browned bits, until the vegetables are soft and pale gold. Stir in the garlic and tomato paste and cook for 1 minute.

Pour in the wine and simmer for 2 minutes, scraping the bottom of the pot. Return the pork and its juices, then add the rosemary, bay leaves, Parmigiano rind, and 1 L of the stock. Bring to a boil, lower to a bare simmer, partly cover, and cook for 30 minutes.

Stir in the rinsed barley and the remaining 500 ml stock. Simmer partly covered for 35 minutes, stirring every 10 minutes, until the barley is tender with a slight chew and the pork breaks apart with a fork. If the pot looks dry, add a splash of water.

Turn off the heat and remove the rosemary, bay leaves, and rind. Cover and rest for 10 minutes so the broth thickens. Stir in the parsley, taste, and add the remaining 1/2 tsp salt if needed. Ladle into bowls with olive oil and grated Parmigiano.