In a large dutch oven, heat oil (2 tablespoons) and sauté onion (1 cup), carrots (3), and celery (2 stalks), stirring occasionally over medium heat, until onion is translucent, about 7-10 minutes. Stir in butter (3 tablespoons) and garlic (4 cloves) and continue to sauté, stirring often so garlic doesn't burn, until fragrant, about 1-2 minutes.
Sprinkle flour (5 tablespoons) over vegetables and stir into a paste. Continue to cook, stir constantly, until the paste begins to become golden.
Slowly whisk in chicken broth (4 cups). Add Worcestershire sauce (2 teaspoons), Better Than Bouillon (1 heaping tablespoon), Cajun seasoning (1 teaspoon), black pepper (1 teaspoon), bay leaf (1), and thyme (2 sprigs).
Add potatoes (3 1/2 cups) and ham (3 cups). Increase heat and bring to a boil. Immediately reduce the heat, adjust to a simmer, and add the milk (2 cups). Cook, stirring occasionally, until potatoes are tender when pierced with a fork. Adjust seasoning and serve.
Notes
Add Toppings: Sprinkle cheddar cheese, crumbled bacon, or green onions on top just before serving.