Melt 2 tablespoons of unsalted butter in a large pot over medium heat, then add 1 cup chopped carrots, 1 diced onion, and 1 cup diced celery. Sauté for 4-5 minutes until tender and fragrant.

Add 3 minced garlic cloves, 1 teaspoon dried parsley, 1 teaspoon dried thyme, and a pinch of salt and pepper. Stir for 1 minute.

Pour in 6 cups of chicken broth, followed by 2 cups of diced ham and 4 cups of cubed Yukon Gold potatoes. Bring to a gentle simmer and cook for about 10 minutes.

In a separate bowl, whisk together 2 tablespoons melted butter and ¼ cup all-purpose flour, then add 1 cup warmed milk gradually until smooth. Microwave on high for 1-2 minutes until thickened and stir into the soup.

Remove the ham bone if used, and shred any remaining meat back into the pot. Stir in 1 cup of shredded cheddar cheese and mix until melted.

Ladle into bowls and garnish with additional cheese and chopped green onions if desired.