Heat the vegetable oil in a large pot or Dutch oven over medium heat. Add the chopped onion and sauté for about 5 minutes, until translucent.
Stir in the minced garlic and grated ginger, cooking for an additional 1-2 minutes until fragrant.
Add the diced carrot, celery, and potato to the pot. Cook for about 5 minutes, stirring occasionally.
If using chicken, add the shredded chicken to the pot (or chickpeas for a vegetarian version). Mix well.
Pour in the coconut milk and chicken broth. Bring the mixture to a simmer.
Add the diced apple, curry powder, turmeric, and cumin. Stir to combine.
Season the stew with salt and pepper to taste. Cover and let it simmer for about 30 minutes, stirring occasionally until the vegetables are tender.
Once the stew is cooked, add the lime juice and stir well.
Serve the Mulligatawny Stew hot, garnished with fresh cilantro if desired.