Heat 1 tablespoon olive oil in a 12-inch nonstick sauté pan or deep skillet over medium-high heat.
Add the diced chicken, then season with 1 1/2 teaspoons chili powder, 1 teaspoon ground cumin, salt, and pepper; cook, tossing occasionally, until chicken is cooked through, about 7 minutes.
Transfer the cooked chicken to a sheet of foil, wrap to keep warm, and set aside.
Return the skillet to medium-high heat and add the remaining 1 1/2 tablespoons olive oil.
Add the diced onion and sauté about 3 minutes until beginning to soften.
Add the chopped bell peppers and continue to sauté 3 more minutes, then add the minced garlic and cook 1 minute more.
Pour in the chicken broth, fire-roasted tomatoes, and black beans; stir in the remaining 1 teaspoon chili powder and 1/2 teaspoon cumin, and season with salt and pepper to taste. Bring the mixture to a boil so it bubbles in the center of the pan.
Add the instant brown rice and toss to coat and submerge the rice in the liquid. Reduce heat to medium, cover, and simmer 6–7 minutes until the rice is tender.
Remove the pan from heat and let it rest, covered, for 5 minutes.
Stir the reserved chicken, lime juice, shredded Mexican cheese, and chopped cilantro into the rice mixture. Serve warm.