In a large stockpot over medium heat, melt the butter, then add the onions, carrots, celery, garlic, and a generous sprinkle of salt, and sauté until the vegetables are tender and onions are translucent, 10 to 15 minutes. Be careful not to brown them.
Add the beer to the vegetables, reduce the heat to medium-low, and simmer for 5 minutes to concentrate it.
In a separate, medium pot, bring the vegetable stock to a simmer.
Add the flour to the beer and vegetable mixture. Stir and cook over medium heat until the mixture thickens and starts to stick to the pot. Gradually pour in the milk, stirring as you do, and then bring the soup to a simmer.
Add the hot vegetable stock to the soup. If your stock cooked down below four cups, add water to get the needed amount.
Remove the pot form the heat and purée with an immersion blender.
Put the puréed soup back onto the stove over low to medium heat, and slowly add the grated cheese, stirring constantly to avoid scorching.
Turn off the burner and stir in the Marash pepper, white pepper, and vinegar. Season with additional salt if needed.
Storage: Store in an airtight container in the refrigerator for up to 5 days.