Sprinkle both sides of the chicken generously with salt. In a large Dutch oven or heavy saucepan, heat the oil over medium heat. Place half of the chicken, skin side down, in the Dutch oven and cook, without moving, until browned, about 8 minutes. Turn the chicken and cook for 2 minutes more; transfer to a plate. Repeat with the remaining chicken.
Meanwhile, in a small saucepan, cover the frying peppers with water and bring to a simmer over medium heat. Cook until tender, about 10 minutes. Transfer the peppers to a blender or food processor with 1/2 cup of the cooking water and blend until smooth. Pour into a bowl.
In the blender, puree 1/2 cup chicken broth with the onion, garlic, black pepper, cumin and 1 teaspoon salt. Scrape into a bowl, then puree the cilantro with 1/2 cup chicken broth. Add the carrots and blend until finely chopped.
Heat the Dutch oven over medium heat, then add the frying-pepper puree. Cook, stirring, until thick, about 4 minutes. Add the onion mixture and cook for 4 minutes longer. Add the cilantro mixture, the remaining 1-1/2 cups broth and the rice and bring to a boil. Add the browned chicken, burying it in the rice. Reduce the heat to medium-low, cover and cook for 15 minutes. Uncover and pour the peas on top. Cover again, turn off the heat and let stand 5 to 10 minutes before serving.