Heat butter in a large pot over medium-high heat.
Add onion, carrot, celery, and a couple large pinches of salt and pepper. Cook, stirring frequently for 8 minutes.
Add garlic, ground mustard, cayenne pepper, and flour, and cook for 1 minute.
Add a couple splashes of broth to deglaze the pan and scrape all the yummy bits off the bottom.
Add potatoes, remaining broth, and a couple large pinches of salt and pepper.
Bring a simmer over medium-high heat.
Reduce heat to low and simmer for 12-15 minutes, stirring every few minutes, until the potatoes and carrots are cooked through and tender.
Remove from heat and stir in cream, shredded cheese (reserve some for garnish), and vinegar until cheese melts.
Season to taste with salt and pepper. Garnish with chives and shredded cheese, and enjoy!
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