Brown the meat: Heat a large pot or Dutch oven over medium-high. Add the ground beef and cook, breaking it up, until browned and no longer pink, about 5–7 minutes.
Drain excess fat if needed.
Sauté aromatics: If the pot looks dry, add olive oil. Stir in onion, celery, and carrots. Cook until softened, 5–6 minutes.
Add garlic and cook 30 seconds until fragrant.
Build the burger base: Stir in tomato paste, smoked paprika, mustard powder (or Dijon), Worcestershire, 1 teaspoon salt, and 1/2 teaspoon pepper. Cook 1 minute to bloom the flavors.
Add potatoes and broth: Toss in the diced potatoes and pour in the broth. Bring to a boil, then reduce to a gentle simmer.
Cook 12–15 minutes, or until potatoes are tender.
Stir in milk: Add the milk and return to a low simmer. Do not boil vigorously after adding dairy.
Thicken and enrich: Remove the pot from heat. In a bowl, whisk the Greek yogurt with a ladle of hot soup to temper it.
Stir the tempered yogurt back into the pot until smooth.
Melt the cheese: Gradually sprinkle in the shredded cheddar while stirring constantly off heat. Keep stirring until completely melted and the soup is silky. Taste and adjust salt and pepper.
Finish and serve: Ladle into bowls.
Top with chopped dill pickles, green onions, diced tomatoes, or a tiny zigzag of ketchup/mustard for that cheeseburger vibe.