Heat vegetable oil in a large skillet over medium heat. Stir in rice; cook and stir until rice is lightly browned and gives off a toasted fragrance, 5 to 7 minutes. Stir in the chicken broth, tomato sauce, green chiles, onion, garlic, salt, cumin, and chili powder. Bring to a boil, then reduce heat to low. Cover and simmer until rice is tender, about 25 minutes, stirring occasionally. Spread Monterey Jack cheese on top, then replace the lid and allow the cheese to melt, about 5 minutes more.
