Heat a large Dutch oven over medium-high heat and melt one tablespoon of butter. Add the chopped onion and saute for about 2-3 mins.
Add 4 more tablespoons of butter. Once melted, mix in the flour and cook it for about one minute.
Whisk in the chicken broth and half and half, making sure you dissolve any lumps that may form. Let this mixture cook for about 2 minutes.
Mix in the broccoli florets, chopped carrots, potatoes, paprika, garlic powder, mustard powder, bay leaf, salt, and pepper to taste.
Bring to a boil, reduce the heat, and simmer for 15 minutes until thick and the veggies are cooked through.
Using an immersion blender, blend the soup to your desired consistency (you can use a regular blender).
Stir in the cheese until melted and check the seasoning adding more salt and pepper, if needed.