Heat the oil in a large pot or Dutch oven over medium heat. Once it's hot, add the ham and cook, stirring often, until the ham has browned up a bit. Then remove it to a seperate plate and cover the plate with foil to keep it warm.
Add the butter to the pot. Once the butter has melted, add the celery, carrots and onion with a big pinch of salt. Cook, stirring often, for 5-7 minutes, until the veggies have begun to soften. Then stir in the garlic and cook until it's fragrant, another minute or so. Increase the heat to med-high, then sprinkle the flour over the veggies and stir to incorporate it. Cook and stir for 2 minutes.
Slowly pour in the broth while stirring, then add the milk and all of the seasonings. Let it all come up to a low boil. Once it begins to boil, stir in the potatoes and return the ham to the pot, reduce the heat back down to medium and let it simmer for about 10 or so minutes, until the potatoes are fork tender. Then remove from the heat and serve.
This soup is also very good with some kielbasa or smoked sausage in place of the ham, pictured below.
