In a large stockpot or Dutch oven, combine the diced potatoes, carrots, celery, onion, and ham. Pour in the chicken broth, making sure the vegetables are covered. Bring to a boil over medium-high heat.

Reduce heat to medium-low, cover, and simmer for 10-15 minutes or until the potatoes are fork-tender.

While the vegetables are cooking, prepare the roux. In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes, stirring constantly until the mixture turns light golden brown.

Gradually whisk in the milk, stirring continuously to prevent lumps from forming. Continue cooking until the mixture thickens slightly, about 3-4 minutes.

Pour the milk mixture into the stockpot with the vegetables and ham. Add the dried parsley and thyme.

Season with salt and pepper to taste. Remember that ham can be salty, so adjust accordingly.

Simmer the soup for an additional 5 minutes, allowing the flavors to meld together. For a thicker soup, you can mash some of the potatoes against the side of the pot.

Ladle the hot soup into bowls and garnish with shredded cheddar cheese and chopped green onions if desired.