Heat oil in a large skillet or cast-iron pan over medium-high heat. Season the chicken pieces with salt and pepper, then add them to the pan and cook until browned.
Add the diced onions and minced garlic to the skillet with the chicken. Cook for 2 to 3 minutes until the onions soften and become fragrant.
Stir in the uncooked white rice, letting it toast slightly in the pan juices for a minute or two.
Pour in the enchilada sauce, diced tomatoes with green chilies, chicken broth, cumin, and oregano. Stir everything well and bring the mixture to a gentle simmer.
Reduce the heat to low, cover the pan with a lid, and cook undisturbed for 15 to 18 minutes until the rice is tender and the liquid is absorbed.
Remove the lid, sprinkle the shredded cheese evenly over the top, cover for 1 to 2 minutes until melted, and serve.