Soak your cashews in hot water for the cheddar sauce. (15 minutes)
Place onion, garlic, carrots and celery in a food processor, and pulse until finely diced. (I do 2 batches: first the onions and garlic, then the celery and carrots. Or, you can just dice by hand.)
In a large pot or Dutch oven, saute the onion, garlic, celery and carrots with about a tablespoon of water on medium high heat until soft, about 5 minutes. Add water a little at a time if things begin to stick.
Add the Dijon mustard and cayenne pepper and stir. Add the potatoes, and stir to heat for about a minute.
Add the broth, beer and water, increase the heat to high, and bring to boil. Once boiling, reduce the heat to medium low, and cook at a rolling simmer, covered, with lid slightly askew to release some steam. Cook for about 10 minutes, or until the potatoes are tender.
While the soup is cooking, make the cheddar sauce. (Pour the sauce into another container if you plan to use the blender to process your soup.)
When the potatoes are tender, turn off the heat. Ladle about ⅓ of the soup at a time into a blender, and process on high until smooth. Don't over blend. Remove the blended portion into a bowl and repeat until all of the soup is blended.
Wipe out any chunks of veggies from the pot and pour the blended soup back in. Place the pot back on the stove on low heat. Add the cheddar sauce and stir to blend in completely. Add sea salt, to taste. Heat to desired temperature and serve.
Garnish with a pinch of smoked paprika, if desired
Notes
It may be tempting, but don't use a dark beer like Guinness or an IPA, as they tend to overpower the soup. I recommend a lighter amber ale or a farmhouse/American cream ale.
To find out if a beer is vegan, you can check the website barnirvore.com