Melt the butter over medium heat in a large stock pot. Add the chopped onion and sauté until translucent. Add the garlic and the diced potatoes. Stir and cook until garlic is fragrant, about 20 seconds.
Carefully, pour the mixture into a large (7-quart) slow cooker. Add the chicken broth, carrots, salt, pepper, paprika, and ground mustard. Stir and then cover the slow cooker with the lid. Cook on low heat for 6 hours OR on high heat for 3 hours. (The potatoes should be fork-tender when cooking time is up.)
Use an immersion blender and blend the soup until smooth. (You can also keep it chunky if you’d prefer.) Pour in the heavy cream, shredded cheddar cheese, and the steamed broccoli florets. Stir to combine. Cover with the lid and allow to heat for another 20 minutes.
Pour soup in individual bowls and serve warm with desired toppings. Enjoy!