Heat 1 tablespoon olive oil in a 12-inch nonstick sauté pan or deep skillet over medium-high heat.
Add the diced chicken, season with 1 1/2 teaspoons chili powder, 1 teaspoon cumin and salt and pepper to taste; cook, tossing occasionally, until no longer pink, about 7 minutes.
Transfer the cooked chicken to a sheet of foil and wrap to keep warm.
Return the skillet to medium-high heat and add the remaining 1 1/2 tablespoons olive oil; add the diced onion and sauté for 3 minutes.
Add the chopped bell peppers and sauté 3 more minutes, then add the minced garlic and cook 1 minute longer.
Pour in the chicken broth, fire-roasted tomatoes, black beans, the remaining 1 teaspoon chili powder and 1/2 teaspoon cumin; season with salt and pepper to taste and bring to a boil so the liquid bubbles in the center.
Add the instant brown rice and toss to coat and submerge in the liquid; reduce heat to medium, cover, and simmer until the rice is tender, about 6–7 minutes.
Remove the pan from heat and let rest, covered, for 5 minutes.
Stir the reserved chicken, lime juice, shredded cheese, and chopped cilantro into the rice until combined and the cheese melts; adjust seasoning and serve warm.