• In a large soup pot, heat oil over medium high heat. Add carrots, onion, and celery, saute until onions are softened, about 4-5 minutes.
• Add salt, garlic, parsley, thyme and pepper and cook 1 minute.
• Add chicken broth, ham, ham bone, and potatoes to the pot. Scrape the bottom with a wooden spoon to remove any browned bits from the bottom.
• Bring to a simmer over medium high heat, then reduce heat to medium low, cover, and cook until potatoes are tender, about 10 minutes.

Thicken the milk:

• Meanwhile, place melted butter in a medium bowl or large glass measuring cup (leave yourself some room at the top as the mixture will bubble up and thicken considerably). Whisk in flour.
• Whisk in milk, using a spatula to get the flour from the sides of the cup as necessary.
• Microwave in 45-60 second intervals (Start with 60, then reduce as the milk gets thicker and closer to being done), whisking well each time, until milk has thickened considerably. Stand at the microwave and keep an eye on it as the mixture will bubble up.

Finish the soup:

• When the potatoes are cooked, remove the ham bone and pull any meat from it to stir back in.
• Stir in the thickened milk and cheddar cheese. Top with additional shredded cheese and green onions as desired.