Add 1 tablespoon butter and olive oil to a large heavy bottomed stock pot/dutch oven over medium heat. Add the diced onion and saute, stirring occasionally, until tender, about 3-5 minutes. Add the minced garlic, paprika, mustard, cayenne pepper, salt and pepper to the pot. Cook for another 45 seconds to 1 minute stirring constantly until fragrant.
Add in remaining 6 tablespoons butter and melt. Once melted, add in the flour and whisk to combine. Cook for 2-4 minutes, whisking constantly, until the flour is thickened.
Gradually add in the chicken stock, whisking constantly. Once all the stock has been added, gradually add in the half and half, again whisking constantly as you add. Reduce heat to medium low and allow to simmer for 15 minutes or until thickened a bit.
Add the chopped broccoli florets and matchstick/shredded carrots to the soup.Allow soup to simmer over low heat for about 15 to 20 minutes, or until broccoli is tender. Stir occasionally.
Add in the cheese and stir until melted. Salt and Pepper to taste.