In a large Dutch oven or heavy-bottomed pot, brown the ground beef over medium-high heat. Break it apart with a spoon as it cooks. Once browned, drain off excess grease and set the cooked beef aside.

Add the diced yellow onion to the pot (with a splash of olive oil if needed, or use residual fat from the beef). Sauté for 5-7 minutes until softened and translucent.

Add the minced garlic and cook for another 1 minute until fragrant. Return the cooked ground beef to the pot.

Push the beef and onion mixture to one side of the pot. Add 2 tablespoons of butter to the empty side and let it melt.

Sprinkle the 1/4 cup of all-purpose flour over the melted butter and whisk continuously for 1-2 minutes to create a roux.

Slowly whisk in the beef broth, ensuring no lumps. Bring the mixture to a simmer, stirring occasionally.

Stir in the whole milk, diced tomatoes (undrained), diced potatoes, carrots, and celery. Add Worcestershire sauce, Dijon mustard, smoked paprika, salt, and black pepper.

Bring the soup to a gentle boil, then reduce heat to low, cover, and simmer for 20-25 minutes, or until the potatoes and carrots are tender. Stir occasionally to prevent sticking.

Once vegetables are tender, reduce heat to the lowest setting. Stir in the softened cream cheese until fully melted and incorporated, ensuring no lumps remain.

Gradually add the freshly shredded sharp cheddar cheese, stirring constantly until it's completely melted and the soup is smooth and creamy. Do not let the soup boil after adding the cheese, as high heat can cause the cheese to separate.

Taste and adjust seasonings as needed.

Ladle the hot cheeseburger soup into bowls. Garnish with optional pickle relish, crumbled bacon, and chopped green onions for that authentic cheeseburger experience.