In a large pot or Dutch oven, melt the butter over medium heat. Add the chopped onion and a pinch of salt. Cook, stirring occasionally, until the onion is soft and translucent, about 3–4 minutes.

Add the minced garlic and cook for another 30 seconds, until fragrant. Sprinkle the flour over the onions and garlic, stirring to create a light roux. Cook for 1–2 minutes to get rid of the raw flour taste.

Slowly pour in the chicken broth, whisking as you go to keep the mixture smooth. Stir in the milk and cream (if using). Bring the mixture to a gentle simmer.

Add the thyme or Italian seasoning and the broccoli florets. Let the soup simmer gently for about 6–8 minutes, or until the broccoli is bright green and just tender.

Stir in the shredded chicken. If you prefer a thicker soup, simmer a few minutes longer uncovered to reduce slightly. Taste and adjust salt and pepper as needed.

Ladle into bowls. If you like, finish with a light grating of Parmesan and another crack of black pepper. Serve with crusty bread or a simple salad.