Cut the bacon into small pieces and place in a saucepan. Cook over medium-low heat until the fat renders and the bacon is cooked. You want the bacon to be cooked, but not crisp. In another saucepan, bring the stock to boil and then pour it over the cooked bacon. Remove the pan from the heat, cover, and let stand for 20 minutes.
While the stock is infusing, toast the bread slices very well, allowing them to burn a little on the edges. Add the toast to the stock, breaking it into pieces if necessary, cover and leave for 10 minutes. During that time the bread will soak up the stock.
Add the hot milk, mustard, and vinegar to the saucepan, then season with salt and pepper. Using an immersion blender, purée the soup until smooth (alternately, transfer the soup to a standard blender to blend). Return the soup to the saucepan and heat gently, stirring to scrape up any browned bits on the bottom of the pan.
When the soup is warm, whisk in the butter, check the seasoning and serve.