Melt 1 tablespoon of butter in a large frying pan. Add ground beef and cook until browned, breaking it up with a spoon. Season lightly with salt and pepper. Set aside.
In a large heavy pot, melt the other 1 tablespoon of butter over medium heat. Add onion, leek, and celery. Cook gently until softened and fragrant, about 5–7 minutes.
Add 1 cup of broth to the aromatics in the pot, then blend with an immersion blender until smooth. This will create a smooth, creamy base for your soup.
Stir in diced potatoes and fresh thyme, then pour in the rest of the chicken broth. Bring to a simmer, cover, and cook until the potatoes are tender, 15–20 minutes.
Take the soup off the heat and remove the thyme sprig.
If your soup isn’t as thick as you’d like, gently mash some of the potatoes in the pot with a potato masher to create a creamier, heartier texture.
Stir in the cream (or half-and-half), half of the cheddar cheese and half of the beef. Warm gently until heated through. Taste and adjust seasoning.
Ladle into bowls and top with some of the reserved ground beef, shredded cheddar, a spoonful of crème fraîche, and a sprinkle of fresh chives.