Heat the oil in a pot over medium heat. Once hot, sauté the celery, carrots, mushrooms, garlic, and onion until soft (about 6 to 8 minutes). Add in the turkey sausage and saute for 2 to 3 minutes more.
Add the chicken broth, potatoes, and kale. Season well with salt and pepper, then bring to a simmer covered with a lid.
Simmer for about 30 minutes. Check the tenderness of the potatoes to make sure they are soft. Taste and adjust the seasoning with salt and pepper if needed.