Heat olive oil and butter together in a large pot over medium heat until melted and hot.
Add diced onion, carrots, and celery to the pot and cook for 7 to 8 minutes, stirring once in a while, until the vegetables soften and release their aromas.
Stir in minced garlic, fresh rosemary, and fresh thyme and cook for 1 to 2 minutes to let the herbs infuse.
Add chicken pieces to the pot and sauté until they are golden brown on all sides and nicely seared.
Sprinkle the flour evenly over the chicken and veggies, stirring well to coat everything and start thickening the base.
Gradually pour in the milk in small amounts while stirring constantly to create a smooth, lump-free mixture.
Add the cubed potatoes and pour in the chicken broth, scraping the bottom of the pot to lift any browned bits for extra flavor.
Bring the soup to a boil, then lower the heat and let it simmer gently for 20 to 25 minutes or until the potatoes are soft when pierced with a fork.
Mix in the shredded cheddar cheese until it has melted fully, creating a creamy texture.
Fold in the chopped fresh parsley and season with salt and freshly ground black pepper to your taste.
Turn off the heat, ladle the soup into bowls, and garnish with additional parsley if desired.
Serve hot and enjoy your rich and hearty chicken and potato soup!