Brown the beef: In a large pot or Dutch oven, heat 1 tablespoon butter or oil over medium-high heat. Add the ground beef, season lightly with salt and pepper, and cook until browned, breaking it up with a spoon. Spoon off excess fat if needed.

Sauté aromatics: Add the onion, carrots, and celery. Cook 5–7 minutes, stirring, until softened. Add the garlic and cook 30 seconds, just until fragrant.

Add seasoning base: Stir in Worcestershire, mustard, paprika, and thyme. Let it toast for 30 seconds to wake up the flavors.

Thicken lightly: Push the mixture to the sides, melt the remaining tablespoon of butter in the center, then sprinkle in the flour. Stir to coat everything. Cook 1 minute to remove the raw flour taste.

Simmer the potatoes: Pour in the beef broth, scraping up any browned bits from the bottom. Add the diced potatoes. Bring to a gentle boil, then reduce to a simmer. Cook 12–15 minutes, until the potatoes are tender when pierced with a fork.

Add dairy: Reduce heat to low. Stir in 1 cup of milk or half-and-half. Do not boil after adding dairy; gentle heat keeps it smooth.

Melt the cheese: Add the shredded cheddar a handful at a time, stirring constantly until melted before adding more. Season with salt and pepper to taste. If you’d like a looser texture, stir in the remaining ½ cup of milk.

Finish and serve: Taste and adjust seasoning. Ladle into bowls and top with green onions or chives. Add chopped pickles and bacon bits if you like that classic burger vibe.