Turn a slow cooker to High. Combine chicken broth, lemon juice, garlic, garlic powder, and cayenne in the bottom of the cooker until mixture heats through, about 5 minutes.
In the meantime, heat oil in a skillet over medium-high heat. Season both sides of chicken thighs with salt and pepper and place in the hot skillet. Cook, turning occasionally, until crisp on both sides, 5 to 7 minutes total. Remove chicken from the pan and set aside.
Turn slow cooker to Low and add rice, carrots, and chicken. Cover and cook until chicken is no longer pink at the bone and the juices run clear, about 4 hours. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).