In a large soup pot, melt the butter over medium heat. Add the onion, celery, and bacon slice and cook until the onion is tender and translucent, about 6 minutes.
Add the flour and cook, stirring constantly, for 2 minutes.
Stir in the wine (if using) or a little of the chicken broth and use a spoon to scrape up the bits on the bottom of the pan.
Add the broth and bring to a boil. Reduce the heat and when the broth stops bubbling, stir in the Cheddar until well combined. Slowly add the cream, stirring constantly. Add the salt and white pepper.
When the cheese has melted and the soup is fully combined, remove and discard the bacon slice. Use an immersion blender to blend the soup until smooth and velvety.
Ladle into bowls or mugs and top each serving with a spoonful of crème fraîche, a scattering of crumbled bacon (if using), and a pinch of fresh chives. Serve hot.
Store leftovers in a covered container in the refrigerator for up to 2 days.
