1. In a Dutch oven or large pot, melt the butter over medium heat. When hot, add the onion and carrot. Cook until soft, about 5 minutes. Add the garlic and cook 30 seconds.
2. Stir in the flour until everything is coated and a roux (paste) is formed. Slowly whisk in the broth then bring to a boil. Once boiling, add the ham. Cook until heated through, about 2-3 minutes. Stir in the milk and cheese until melted. Garnish with parsley before serving.
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