Temper the whole spices: Heat the coconut oil in a large heavy pot or Dutch oven over medium heat. Add the cardamom pods, cloves, cinnamon stick and crushed black peppercorns, and saute for 30 seconds until fragrant. Add the curry leaves and slit green chilies - they'll splutter, so stand back slightly.
Soften the aromatics: Add the sliced onions and cook for 6-7 minutes until soft and translucent but NOT browned. Stir in the ginger garlic paste and cook 1 minute until the raw smell disappears.
Add chicken and thin coconut milk: Add the chicken pieces and salt, tossing to coat in the spiced oil. Pour in the thin coconut milk. Add the potatoes and carrots, bring to a gentle simmer, then cover and cook on low for 20-25 minutes until the chicken is cooked through and the vegetables are fork-tender.
Finish with thick coconut milk: Uncover and stir in the thick coconut milk. Simmer gently for 3-4 minutes - do NOT let it come to a rolling boil, or the coconut milk may split. Taste and adjust salt.
Temper and serve: In a small pan, heat the remaining teaspoon of coconut oil and fry the curry leaves and crushed peppercorns for 20 seconds until crisp. Pour this tempering over the stew. Serve hot with appams, idiyappam, or bread.