Fry bacon pieces in a small saucepan over medium heat until they start to take on color. Add chicken stock and bring to a boil. Remove from heat, cover, and infuse for 1 hour.
Place the thin slice of bacon on a parchment-lined baking sheet. Cover the bacon with an additional piece of parchment and then with a couple more baking sheets. It may be necessary to invert the bottom sheet so the bacon is squeezed flat. Bake until the bacon is brown and crisp, 10 to 15 minutes. Set the cooked bacon aside on a cooling rack until needed.
Bake the bread slices until brown on both sides, about 5 to 7 minutes per side. Set aside until needed.
Wash the eggs in their shells and set aside.
When the stock has infused for an hour, strain out the bacon pieces and discard. Combine the stock with the milk and bread in a blender. Puree until smooth, about 2 minutes.
Strain the puree into a saucepan, add the vinegar and mustard, and bring to a boil. Remove from heat and stir in the butter. Salt and pepper to taste.
Divide the soup among serving bowls. Carefully break each egg. Put the yolk in half of the shell and carefully place in the center of the soup. Reserve the white for other purposes. Decorate each serving with half a bacon slice and half the chives.