Cook the Ground Beef: In a spacious saucepan, warm the vegetable oil over medium heat. Incorporate the ground beef along with 1 teaspoon of salt and black pepper. Sauté the beef for about 5-7 minutes, stirring with a spoon to break it apart until it’s nicely browned. Once fully cooked, transfer the beef to a plate and set it aside.

Sauté the Onion: In the same pot, add the chopped white onion and sauté for 4-5 minutes until softened and translucent. Add the flour and stir well to combine, cooking for 1-2 minutes to remove the raw flour taste.

Add the Broth and Potatoes: Gradually pour in the low-sodium chicken broth, whisking continuously to ensure a smooth texture. Add the quartered potatoes and bring the mixture to a simmer. Cook for about 10-12 minutes, or until the potatoes are tender.

Melt the Cheese: Lower the heat and stir in 1 cup of the shredded American cheese, dry mustard, Worcestershire sauce, and the remaining teaspoon of salt.

Reintroduce the Beef: Add the cooked ground beef back into the soup. Stir everything together and let it simmer on low heat for an additional 5 minutes to allow the flavors to meld.

Prepare the Bun Topping: In a small pan, melt the butter and toast the hamburger bun until golden brown on both sides. Cut the bun into small pieces and set aside.

Assemble and Serve: Ladle the cheeseburger soup into bowls. Garnish each bowl with the remaining shredded American cheese, crumbled bacon, chopped green onions, and a dollop of dill pickle relish. Add the toasted hamburger bun pieces on top for a delightful crunch.