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In a large, deep-sided skillet over medium-high heat, heat oil. Season chicken all over with salt and pepper. Arrange chicken in skillet skin side down and cook, turning occasionally, until browned, about 5 minutes per side. Transfer chicken to a plate.
Reduce heat to medium. In same skillet, cook onion, stirring occasionally, until softened, about 5 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Add rice and cook, stirring frequently, until slightly toasted and fragrant, about 2 minutes. Add chiles and cook, scraping up any browned bits on bottom of pan. Add broth, oregano, and cumin; season with salt and pepper. Return chicken to skillet and bring to a boil, then cover and reduce heat to low. Simmer, stirring occasionally, until rice is cooked and chicken is cooked through, 12 to 15 minutes. Transfer chicken to a clean plate.
Add cheese and sour cream to skillet and stir until melted. Stir in cilantro.
Return chicken to skillet. Serve with lime wedges alongside.