Melt the butter in a large pot over medium heat. Sauté the onions and celery in the butter for 6-7 minutes, or until tender. Add garlic and cook another 1-2 minutes.
Stir in the water and potatoes, bring to a boil and then simmer until the potatoes are tender. Add the cheese and stir until melted.
In a separate small bowl, combine the flour and milk and stir well until the flour is mostly dissolved. Slowly add mixture to the soup, stirring constantly, until soup has thickened, about 5 minutes.
If you would like a thicker soup, transfer a few cups of soup to a blender (or use an immersion blender) and carefully puree the soup until it reaches a creamier consistency. Return pureed soup to pot and combine. For a chunkier soup, skip this step.
Toast 6 slices Martin’s Butter Bread. Using a shamrock-shaped cookie cutter, cut shamrock shapes out of each toast slice.
Ladle soup into bowls. Top with shamrock toast and extra cheese; garnish with parsley or chives, and/or paprika.