Combine the chicken, potatoes, 1/2 teaspoon of the turmeric, 1/4 teaspoon of the cayenne pepper, and 1/4 teaspoon of salt in a large bowl and stir to coat the chicken and potatoes with the spices. Let stand while you make the sauce.
Finely mince the onion, garlic, and ginger in a food processor and set aside. Combine 2 tablespoons of the oil, the cinnamon stick, cardamom, cloves, black peppercorns, red chiles, coriander, and cumin in a large casserole or pot over medium-high heat. Cook, stirring, until fragrant and the cinnamon unfurls, 1 to 2 minutes.
Add the minced onion mixture, the green chile, and 1 teaspoon of salt and cook, stirring, until the vegetables begin to brown around the edges, 10 to 15 minutes. You may need to turn the heat down to medium if they begin to brown too quickly.
Remove and discard the cinnamon stick and green chile and stir in the remaining 1/4 teaspoon of turmeric and 1/4 teaspoon of cayenne. Add the tomatoes and tomato paste and cook, stirring, for 5 minutes. Let cool slightly, then pour into a food processor or blender and purée until smooth. Set aside.
Heat the remaining 1 tablespoon of the oil in the same casserole or pot over medium-high heat. Add the chicken and potatoes and cook, stirring, for 2 minutes. Add the yogurt, 1 tablespoon at a time, and stir well after each addition. Cook, stirring, to evaporate some of the moisture, about 2 minutes.
Add the puréed tomato mixture and bring to a boil. Stir in the water. Return to a boil, then reduce the heat and simmer, partially covered, until the chicken is cooked through, about 30 minutes, stirring every 5 to 8 minutes and scraping the bottom of the pan to keep the sauce from sticking. Uncover and cook until the sauce is reduced and thickened somewhat, 5 minutes more. Stir in the cilantro and lemon juice. Season with salt to taste. Serve hot.