Heat olive oil in a large skillet over medium-high heat. Season skin side of chicken with salt and pepper. Place in the hot skillet, skin-side down. Cook until golden brown but not burned, about 5 minutes. Flip, reduce heat to medium, and sear until golden brown on the other side, about 2 minutes. Remove chicken from skillet.
Combine chicken broth, rice, water, garlic, and 1/4 teaspoon salt in the same skillet. Bring to a boil and mix well. Place the chicken on top, forming wells in the rice. Reduce heat to low. Cover with a lid and simmer until rice is tender and chicken is no longer pink in the center, 15 to 20 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).
Remove chicken from skillet. Mix in black beans, cilantro, and lime juice. Serve together.
