In a large soup pot, heat a tablespoon of oil over medium-high heat. Add the chopped carrots, onion, and celery, sautéing for about 4-5 minutes until softened.

Stir in the minced garlic and seasonings. Cook for an additional minute, allowing the garlic to bloom.

Pour in the chicken broth and add the diced ham and Yukon Gold potatoes. Bring to a gentle simmer and cook for about 10 minutes.

In a small bowl, melt butter and whisk in the flour until smooth. Gradually stir in warm milk to create a creamy mixture.

Microwave the milk mixture for 45-60 seconds, whisking until it thickens. Stir into the soup to achieve a creamy consistency.

Remove any ham bone and stir in the shredded cheddar cheese until melted. Heat through on low for a few more minutes.

Ladle the warm soup into bowls, topping with additional cheese and green onions if desired. Serve with crusty bread.