Heat a skillet to medium heat and add olive oil. Once hot, add the garlic and leeks. Cook for 3-4 minutes.
Place the cooked garlic and leeks into a slow cooker with the potatoes, carrots, celery, ham, rosemary, black pepper and vegetable broth.
Place the lid on the slow cooker and cook on LOW for 6-8 hours or HIGH for 3-4 hours.
Once finished, stir in the milk and cook for another 30-60 minutes.
To thicken, scoop about 1 cup of potatoes with some broth into a food processor. Blend until smooth, then stir back into the soup.
