To a large pot over medium heat, saute the shallots, carrot, celery, and fresh rosemary (or substitute) in the olive oil and butter. Season with half of the salt and a couple cracks of freshly ground black pepper. Cook, stirring occasionally, for about 5 minutes or until the onions are translucent and the carrots are starting to soften.
Add the cubed potatoes, broth, the other ½ teaspoon of salt, and dairy-free milk. Bring to a boil and then reduce the heat to a simmer. Cook for about 8-10 minutes or until the potatoes can easily be pierced with a fork.
Pull off the heat, add a splash of vinegar, and using an immersion blender, blend until it's completely smooth or set aside a cup of the cooked potatoes and then add them back in once blended for some texture. Taste and add more salt, pepper, and/or vinegar as needed.
Add any toppings you'd like or enjoy as is.