In a large stock pot, heat olive oil and sauté onion until tender, about 5 minutes. Add garlic and cook for 2 minutes. Add diced potatoes and broth to the pot. Bring to boil and reduce to medium heat. Let potatoes cook until tender, about 15 minutes. With a slotted spoon, remove 1 1/2 cups of the potatoes. Set aside. Puree the remaining potatoes with a hand blender until smooth.

In a medium saucepan, melt butter. Stir in flour and cook for 2 minutes. Whisk in milk, bring to a boil and simmer until thickened. Slowly stir mixture into large stock pot with the pureed potatoes. Stir the reserved potatoes back into the soup.

Stir in the shredded cheese until melted. Add in the diced ham. Season with salt and pepper, to taste. Ladle the soup into bowls and garnish with green onion, if desired. Enjoy!