For the toasted sesame buns: Set the oven to broil. Toss the sesame bun chunks on a sheet pan with the oil and a pinch of salt and pepper. Broil until golden brown, about 5 minutes.
For the cheeseburger potato soup: In a large Dutch oven, cook the bacon over medium heat until crispy, about 7 minutes. Remove to a paper towel lined plate to cool, then roughly chop. Set aside.
Add the beef and a pinch of salt to the Dutch oven and cook, breaking up with a spoon, until browned, 4 to 5 minutes (it will continue to cook through). Add the onion, carrot, celery and a good pinch of salt and pepper and cook until translucent, about 10 minutes. Add the garlic and mustard powder and cook for another minute. Add the potatoes, stock, parsley, 1 1/2 teaspoons salt and a bunch of turns of pepper and bring to a boil, then simmer until the potatoes are tender (pierce with a fork to check), 30 to 40 minutes.
Add the heavy cream and 8 ounces cheese and stir until the cheese is melted.
Finish the soup with the pickle brine. Taste and adjust for seasoning as desired. Ladle the soup into bowls, then top with some of the remaining shredded cheese, pickles, bacon, a drizzle of ketchup, mayo if you're freaky, and toasted sesame buns.