In a large pot, heat 1 tablespoon of olive oil over medium heat. Add the sliced onion and minced garlic, sauté for 2-3 minutes until fragrant.
Add diced chicken to the sautéed onion and cook for about 5 minutes, just until no longer pink. Be careful not to overcook, as the chicken will simmer further in the soup.
Add diced carrots and broccoli florets to the pot, then sauté for 2-3 minutes with a tablespoon of water to prevent sticking.
Add kale and vegetable broth. Give it a stir and bring to a boil, then lower the heat and let simmer, covered, for about 10 minutes. Stir frequently.
Stir in the chickpeas, the milk, salt and pepper to taste. Simmer for additional 5 minutes, then add the remaining 2 tablespoons of olive oil. Taste and adjust with more seasonings, if necessary.
Serve warm, with a sprinkle of Parmesan cheese and a drizzle of olive oil if desired.