In a dutch oven, heat the olive oil over medium-high heat. Season the chicken with salt and pepper and cook in the pot until golden on both sides, 3 to 4 minutes; transfer to a work surface. Add three-quarters of the onion to the pot, lower the heat to medium and cook, stirring occasionally, until just beginning to brown, 2 to 3 minutes. Add the potatoes and cook for 1 minute.
Cut the chicken into bite-size pieces and return to the pot with any of its juices. Stir in the rice, chicken broth and 3 cups water and bring to a boil. Lower the heat and simmer, stirring occasionally, until the potatoes are tender, 15 to 18 minutes; season with salt and pepper. Stir in the parsley and avocado. Serve with the sour cream and remaining onion.