Heat the EVOO in a Dutch oven over medium-high heat and melt the butter into the oil. Add the garlic, onions, 1 apple and the ginger and season with salt and pepper. Cook until tender, 7 to 8 minutes. Add the coriander, cumin, turmeric and ground mustard and cook, stirring, until fragrant. Add the flour and cook, stirring, for 1 minute. Add the stock and simmer until thickened to the consistency of a light sauce, about 20 minutes. Stir in the chicken.
Put the remaining apple in a food processor with the yogurt, cilantro and mint. Pulse to combine. Add the cardamom and the juice of 1 lime and season with salt and pepper. Whizz to combine. Refrigerate until ready to serve.
To serve, prepare the rice according to the package directions. Cut the remaining lime into wedges. Heat a griddle or grill pan over medium-high heat and splash some water on it, then grill the naan and brush with melted butter. Ladle the mulligatawny into shallow bowls and top with the rice and a dollop of raita. Serve the naan and lime wedges on the side.
Put the chicken in a very large stockpot or divide between 2 pots if necessary. Add the peppercorns, garlic, bay leaves, carrots, celery, lemons, onions and herb bundle and sprinkle with salt. Add enough water to cover the chicken. Bring to a boil, then reduce the heat to low and cook at a rolling simmer for 1 1/2 hours. Remove the chicken from the liquid and let cool. Strain the stock. (You should have 5 to 6 quarts stock.) Remove the skin and bones from the chicken, and shred the meat using your hands or 2 forks.
The mulligatawny can be covered and refrigerated for a make-ahead meal. Reheat over medium heat. The apple-yogurt raita can be stored in the refrigerator. Cook the rice just before serving.