Wash and peel the Yukon Gold potatoes, then dice into uniform chunks. Chop the onion, carrots, and celery, and mince the garlic.

Heat olive oil in a large pot over medium heat. Add the onion and sauté for 5-7 minutes until softened. Mix in the carrots, celery, and garlic, cooking for another 3-4 minutes.

Add the diced potatoes, vegetable broth, thyme, rosemary, salt, and pepper. Bring to a gentle boil, then reduce to low and simmer uncovered for 20-25 minutes.

Use an immersion blender to purée the soup until smooth. Reserve some chunky soup mixture for texture and stir back in. Incorporate heavy cream and milk, heating gently.

Ladle into bowls, garnish with chives and cheddar cheese, and serve immediately with crusty bread.