1. Using a sharp knife, trim the green portions of the leek and halve the white part of the leek lengthwise. Rinse well under cold running water to rid the leek of any sand. Slice thinly crosswise and set aside.
2. In a large soup pot over medium heat, melt the butter and add the bacon. Cook for 5 to 6 minutes, stirring occasionally, until the bacon has rendered most of its fat. Add the chopped leeks and cook until wilted, about 5 minutes. Add the chicken broth, potatoes, salt, black pepper and all of the herbs and bring to a boil. Reduce the heat to a simmer and cook for 30 minutes, or until the potatoes are falling apart and the soup is very flavorful.
3. Fish out the thyme twigs and the bay leaves and, working in batches, puree the soup in a food processor or blender. (Use an immersion blender directly in the pot if you have one.) Adjust the seasoning, if necessary. Serve immediately, with some of the chives sprinkled over the top of each bowl of soup.