Start the night before by soaking your lentils in one bowl with water and a splash of apple cider vinegar or a squeeze of lemon juice. Soak your barley in another bowl also with a bit of apple cider vinegar or lemon juice.
The next day, drain the lentils and barley and rinse each separately very well in a colander under cool running water. Set aside.
Warm the fat (olive oil or butter) in a large soup pot over medium heat.
Add onion, celery, and carrots. Sauté until softened, about 5–7 minutes.
Stir in the garlic and cook for 1 minute more, until fragrant.
Add soaked lentils, barley, broth or water, bay leaf, salt, and any optional flavorings (like tomato paste or Parmesan rind).
Bring to a gentle boil, then reduce heat to low. Cover partially and simmer for 45–60 minutes, stirring occasionally, until lentils and barley are tender.
Taste and adjust seasoning with additional salt and black pepper. Remove the bay leaf.
If using greens, stir them in during the last 10 minutes of cooking.
Serve warm with a drizzle of olive oil and, if desired, a grating of Parmigiano-Reggiano on top. Or best of all, top the soup with a dollop of Crema di Latte.