Preheat the oven to 325 degrees. Lay a large piece of aluminum foil on the counter.
Combine the salt, pepper, cumin, chili powder and oregano in a small bowl.
Arrange the chicken on the foil in a single layer. Use half the seasoning mixture to sprinkle the tenderloins evenly on both sides.
Cut one of the limes in half, squeezing all of its juice evenly over the chicken. Sprinkle with 1 tablespoon of the oil. Rub the chicken on both sides to evenly distribute the spices, juice and oil. Let sit for 10 minutes.
Heat the remaining tablespoon of oil in a 4-quart shallow braising pan, ovenproof skillet or Dutch oven over medium-high heat. Add half of the spice-rubbed chicken; cook, turning once, until lightly browned on both sides. It's not necessary to cook comletely through. Transfer to a plate; repeat with the remaining chicken.
Reduce the heat to medium. Add the onion, bell pepper and a pinch of salt to the pan, plus more oil if needed; cook, stirring, for 4 minutes or until the vegetables have softened. Add the tomato paste and the remaining spice mixture; cook, stirring to combine, for 1 minute. Add the rice and stir to incorporate. Add the broth; increase the heat to medium-high and stir to combine; cook until the broth begins to boil.
Cover and transfer to the oven. Bake for 20 minutes, then remove from the oven and arrange the cooked chicken tenders over the rice. Cover and bake for 8 minutes, then check to see whether the rice has cooked through and all of the broth has been absorbed; if not, bake for 5 minutes or until rice has cooked through. Remove from the oven, uncover and sprinkle with cheese and cilantro.
Serve right from the cooking vessel at the table, shielding the hot handles with pot holders. Cut the remaining lime into wedges to serve on the side.