Bring a medium pot or large frying pan to medium high heat and add a splash of olive oil. Add the sausage and cook for 4 minutes to lightly brown on some of the edges, breaking up the sausage as it cooks.
Add in the onion, carrot, and celery and cook another 5-8 minutes until the veggies soften. Turn heat to medium low.
Add in the butter and garlic and cook for 3 minutes. Stir in the flour, stirring constantly to make sure the flour is incorporated and there are no lumps.
Whisk in the broth, continuing to whisk until it comes to a simmer. You will notice it thicken slightly. Cook at a light simmer for about 15 minutes.
Stir in the milk and bring to a simmer. Remove from heat. Stir in the mustard.
This is the hardest step! Make sure the sauce is not simmering at all, and start adding the cheese. Add about 1/4 of the cheese and stir, allowing it to slowly melt and incorporate. Continue only after all the cheese is melted. If it feels like this is going too slow, you can turn the heat super low, but be careful! If you see melted cheese clumping at the bottom, immediately remove from the heat and continue to whisk as it cools.
Once all the cheese is incorporated, you can bring the heat back on super low to warm up the soup so it is ready to serve.
Meanwhile, cook the pasta. Strain and split between the bowls.
Ladle the soup over the pasta and serve. Top with parsley, black pepper, and hot sauce.