In a large pot, combine butter and olive oil over medium heat. Add diced onions, carrots, and celery. Sauté for about 7-8 minutes until softened and onions are translucent.

Stir in minced garlic, dried rosemary, and thyme. Cook for an additional 1-2 minutes until fragrant.

Add cubed chicken breast and cook for about 5-7 minutes until golden on all sides.

Sprinkle flour over the mixture and stir for 1-2 minutes to incorporate.

Gradually pour in milk while stirring to prevent lumps, then stir for another 2-3 minutes until thickened.

Add cubed Yukon Gold potatoes and chicken stock. Bring to a gentle boil.

Reduce heat to low, cover, and simmer for 20-25 minutes until potatoes are tender.

Stir in shredded cheddar cheese until melted, then sprinkle with parsley.

Serve the soup warm, garnished with additional parsley.