Heat a large heavy bottomed pot over medium. Add enough oil or butter just to cover the bottom of the pan.

Add the onion and cook until it starts to turn translucent.

Add the potatoes and carrots and cook until almost tender, stirring occasionally.

Add the cornstarch and stir and cook for 3-5 minutes. (This cooks the starchy, floury taste out.)

Add the ham, salt, parsley, milk, and chicken broth and bring to a simmer.

Let simmer until thickened, about 2-4 minutes

Removed from heat and add 3/4 of the cheese and stir until the cheese is melted.

Stir in the half and half, then serve with the remaining cheddar cheese.