Preheat the oven to 375 degrees F. Pat the chicken dry and season with salt and pepper. Melt the butter with the oil in a large deep skillet, with a lid over medium-high heat. Add the chicken, skin-side down, and cook until well browned, about 6 minutes. Turn chicken and cook until other side browns lightly, about 2 minutes more. Transfer chicken to a plate. Pour off all but 2 tablespoons fat.
Add the onion to the skillet and cook, stirring often, until soft, 3 to 5 minutes. Add the garlic and cook 1 minute longer. Add the peppers and cook, stirring, until softened, about 3 minutes. Stir in the turmeric, cumin, rice, and 1/4 teaspoon salt and stir until the rice is coated. Pour in the broth, stir to distribute the rice evenly. Increase the heat to high and boil until liquid begins to be absorbed into the rice, about 2 minutes. Nestle the chicken in the rice. Cover and transfer the pan to the oven and bake until the rice is tender and chicken cooks through, about 25 minutes.
Remove the cover, fluff the rice and stir in the peas. Cover and let sit to warm the peas, about 5 minutes. Season with additional salt and pepper. Serve.
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