Heat a large heavy-bottomed pot over medium heat and melt the butter until it starts to foam and smell nutty.
Add the finely chopped onion to the pot and sauté until translucent and fragrant, about 5 minutes. Stir occasionally to prevent sticking.
Stir in the diced celery and carrots, cooking for another 5 minutes until they begin to soften and release their sweet aroma.
Pour in the chicken broth and bring it to a gentle simmer, then add the shredded cooked chicken. Let it simmer for about 15 minutes to meld all the flavors and heat the chicken through.
Stir in the cream and cook for another 2-3 minutes, allowing the soup to become velvety and rich, with a slight thickening.
Taste the soup and adjust seasoning with salt and pepper if needed. Sprinkle chopped parsley over the top for a fresh, bright finish.
Ladle the hot, creamy soup into bowls and serve immediately, enjoying the comforting aroma and smooth texture.