Combine first 8 ingredients in a large Dutch oven over medium-high heat; bring to a boil. Add chicken; return to a boil. Cover, reduce heat, and simmer 10 minutes, stirring occasionally.
Add potatoes; return to a simmer. Cover and cook 25 minutes or until potatoes are tender, stirring occasionally. Stir in cilantro and pepper.
Heat oils in another Dutch oven or large skillet over medium-high heat. Add half of spinach; cook 1 minute, tossing constantly. Add remaining spinach; cook 2 minutes or until spinach wilts, tossing constantly. Serve chicken mixture over spinach.