Wash the turtle beans and put them in a saucepan along with the stock and 1 tablespoon of oil. Cover tightly, bring to a boil, and simmer for 2½ hours or so, until beans are quite tender.
Chop the onion and sauté in the remaining oil until soft. Chop the celery, including the leaves. Grate potato and carrot on large grater. Add celery, potato, and carrot to onion and cook over medium heat for several minutes, stirring all the while.
Add the vegetables to the beans, along with the seasonings, in the last hour of their cooking. Include garlic if desired.
Bring the soup to a boil and lower the heat to simmer until the beans and vegetables are done.
Add the lemon juice and lemon slices when the soup has finished cooking.