Season both sides of the chicken with salt and pepper. Set a 12-inch skillet over medium-high heat and add the oil. When it starts to shimmer, add the chicken and cook on one side until golden brown, about 4 minutes. Transfer the chicken to a plate, leaving the fat in the pan.
Add the rice, onion and 1/2 teaspoon salt to the pan. Cook, stirring frequently, until the rice is toasted and the onions begin to soften, about 3 minutes. Add the garlic, oregano, chili powder and cayenne, cooking just until fragrant, about 15 seconds.
Mix in the tomatoes, green pepper and chicken broth. Bring the mixture to a boil, scraping up any browned bits from the bottom of the pan. Add the chicken back to the pan, browned side up, along with any accumulated juices. Cover the pan and reduce the heat to low. Cook until the chicken is cooked through, about 12 minutes, depending on size of the breasts.
Transfer the chicken to a cutting board and gently give the rice a quick stir then replace the cover and continue cooking until the rice is tender and the liquid is absorbed, about 12 minutes. Meanwhile, cut the chicken into 1/2-inch slices.
Turn off the heat under the pan then arrange the sliced chicken on top. Sprinkle with the cheese and green onions then cover the pan again and let stand for a few minutes until the cheese has melted.