Preheat oven to 375°F, and prepare the peppers by cutting them in half lengthwise and removing the seeds and membrane; set aside in a baking dish cut side up.
Heat olive oil in a large skillet over medium heat. Add onions and saute until they start to soften, about 2-3 minutes.
Add garlic, chicken, chili powder, cumin and season with salt and pepper. Cook, stirring occasionally, until the chicken is cooked through, about 5-7 minutes.
Add rice, cilantro, lime juice, and stir together until heated through, about 2 minutes.
Spoon the cooked chicken and rice mixture into the cut pepper halves, filling them to the top. Pour water into the baking dish to fill the bottom, being careful not to pour any water on the peppers.
Cover baking dish with foil and bake until peppers are softened, about 20 minutes. Remove from oven, sprinkle with shredded cheese and return to oven to bake until cheese is melted, about 5 more minutes
Serve immediately, topped with pico de gallo and sour cream, if desired.