Brown the bacon in a large heavy pot (on medium low) and remove. Pat with bacon with paper towels and cut into pieces. Save some some bacon fat. Add the onions to the pot.
Brown the onion in the bacon fat (about five minutes). Add a little olive oil and the green beans and saute (on medium low ) for ten minutes or until onions are translucent.
Add the broth and bring to a boil. Add potatoes and simmer( on medium heat) until the potatoes and green beans are cooked through and slightly soft( about fifteen minutes).
Meanwhile, make the roux. Add the butter to a small saucepan and melt on medium heat then add the flour. Cook and stir until the roux is golden brown (about five minutes). Add the roux to the soup. Turn the heat off and add the milk to the saucepan and stir, combining the milk with any remaining roux that is stuck to the pan and until the milk is slightly warm. Add to the soup.
Simmer the soup for about five minutes, season with salt and pepper. Add the bacon to the soup or top individual servings with bacon. Note: This soup is thick and chunky...add some extra milk, broth or water if a thinner consistency is desired.
