Sauté the vegetables in butter until softened. In a large pot over medium heat, melt the butter and add the diced onion, celery, and carrots. Cook for about 5 minutes, stirring frequently until everything is softened and fragrant.

Add garlic and flour, stirring briefly. Toss in the minced garlic and sprinkle the all-purpose flour. Cook for another minute, ensuring the garlic is fragrant but not browned.

Slowly pour in the broth, stirring until smooth. Gradually add the 5 cups of chicken broth while stirring continuously until the mixture is smooth and homogenous.

Incorporate the potatoes and simmer until tender. Add the cubed potatoes and bring the mixture to a gentle boil. Simmer for about 20 minutes or until the potatoes are fork-tender.

Stir in the chicken and cook until done. Fold in the diced chicken and let it cook for about 10-15 minutes until the chicken is fully cooked and tender.

Add milk or cream, season, and simmer gently. Finally, pour in the milk or cream, and stir well. Taste for seasoning and adjust with salt and pepper if needed. Let it simmer for an additional 5 minutes.