In a large soup pot over the stove top on medium-high heat, add olive oil and minced garlic. Add diced ham and fry, while stirring, until ham starts to brown a little.
Add chicken broth, potatoes, and broccoli and allow it to come to a boil.
Cover the pot and allow it to simmer until the potatoes are tender. Stir occasionally.
Whisk together heavy cream and flour in a bowl.
Over low heat, once the potatoes are tender, slowly stir in the heavy cream mixture. Add the cheddar cheese and stir until melted.
Stir soup over low heat until it simmers and thickens. This won't take long.
Add rosemary, oregano, sage, thyme, black pepper and salt (as needed) to taste. Stir well and serve.