Heat oil in a Dutch oven over medium heat. Add onion and cook, stirring, until softened, about 3 minutes. Add beer; bring to a boil and boil for 5 minutes. Add potatoes, broth and water; cover and bring to a boil. Reduce the heat to maintain a simmer, and cook until the potatoes are tender, about 4 minutes. Remove from the heat and mash the potatoes with a potato masher to the desired consistency.

Whisk milk and flour and add to the soup. Bring to a simmer over medium-high heat and cook, whisking occasionally, until thickened, about 3 minutes. Remove from the heat; stir in 1 1/4 cups cheese and stir until melted. Ladle the soup into bowls and garnish with the remaining 1/4 cup cheese and bell pepper.