In a 3 1/2 quart Dutch oven over medium heat, add butter, oil, onion, and 1/2 tsp salt. Cook covered for 8 minutes, stirring occasionally. Uncover and add Cajun seasoning, garlic, thyme, smoked paprika and continue cooking until the onions are tender. About another 5 minutes.

Add the Vermouth and let it come to a simmer. Pour in the chicken broth and milk, stir constantly. Bring to a boil. Stir in the broccoli and reduce the heat until it simmers.

Use an immersion blender and blend to desired consistency.

Add 3 cups of shredded cheese and stir until incorporated.

Salt and pepper to taste.

Use the remaining cheese and parsley to garnish.

Serve.