Brown the sausage. In a large pot, cook the Italian sausage until browned and fully cooked. Remove it from the pot and set it aside.
Cook the vegetables. Add the butter to the same pot and let it melt. Add the carrots and celery, then cook for 6–8 minutes until softened. Stir in the diced onion and cook for another 2–3 minutes.
Add the garlic and flour. Stir in the minced garlic and cook for 1 minute. Add the flour and stir well, cooking for another 1–2 minutes.
Pour in the broth. Add a splash of chicken broth and stir to loosen the mixture. Slowly add the remaining broth, stirring until everything is well combined.
Add the potatoes and seasonings. Stir in the cubed red potatoes, parsley, Italian seasoning, ground mustard, pepper, and hot sauce. Bring the soup to a boil, then reduce the heat and let it simmer for 20–25 minutes, or until the potatoes are fork-tender.
Make it creamy. Return the cooked sausage to the pot. Stir in the heavy cream and shredded cheddar cheese until the cheese melts and everything is well combined.
Serve. Ladle the soup into bowls and serve warm with your favorite bread or salad.