In a large soup pot, melt the butter over medium heat. Add the onion, celery, and carrot. Sauté for about 5–7 minutes until softened. Add garlic and cook another minute.
Pour in the chicken broth and beer. Bring to a simmer for 10 minutes to let the alcohol cook off and the flavors blend.
Add almond milk, Dijon mustard, paprika, salt, and pepper. Stir in the cream cheese until smooth.
Mix the arrowroot with water, then whisk it into the soup to thicken. Simmer for 3-4 minutes.
Lower the heat and slowly whisk in the shredded cheddar cheese until smooth and creamy. Don't boil once the cheese is added, this keeps the texture silky.
Taste and adjust seasonings. Serve hot with your favorite toppings!