Brown the beef: Heat a large pot or Dutch oven over medium-high heat.
Add the ground beef (and olive oil if the pan is dry). Cook, breaking it up, until browned with crisp edges, 5–7 minutes. Season lightly with salt and pepper, then use a spoon to remove excess fat if needed.
Sweat the aromatics: Add the onion, carrots, and celery to the pot.
Cook, stirring, until the onion is translucent and the veggies soften slightly, about 5 minutes. Stir in the garlic and cook 30 seconds until fragrant.
Build flavor: Stir in tomato paste, smoked paprika, and dry mustard. Cook 1 minute to toast the spices.
Add Worcestershire sauce and scrape up any browned bits from the bottom of the pot.
Add potatoes and broth: Stir in the diced potatoes and pour in the broth. Bring to a boil, then reduce to a simmer. Cover and cook until the potatoes are tender when pierced, about 12–15 minutes.
Make it creamy: Reduce the heat to low.
Stir in the milk and cream cheese until the cream cheese melts and the broth looks slightly thickened and smooth.
Melt the cheddar: Sprinkle in the shredded cheddar a handful at a time, stirring constantly so it melts evenly. Keep the heat low and do not boil—this prevents the cheese from separating.
Finish with yogurt: Whisk the Greek yogurt in a small bowl with a ladle of hot soup to temper it. Stir the mixture back into the pot.
Taste and adjust salt and pepper. If you want it thicker, simmer uncovered for 3–5 minutes more, stirring often.
Serve and top: Ladle into bowls and add your favorite burger-inspired toppings. Chopped pickles and a few sesame seeds give it that classic cheeseburger vibe.