Place all ingredients for chili in slow cooker. Stir to combine. Cover with lid and cook on low for 6 hours.
At 5½ hours, remove chicken from pot and chop into small pieces. Return chicken to pot and replace lid.
In a sauce pan over medium heat, add oil and rice. Stir and cook until rice is lightly toasted. Add tomato sauce and cumin, stir to combine. Add chicken broth and cover with lid. Cook for 15 minutes (do not remove lid).
Place a portion of rice in a shallow bowl. Ladle chili over top and garnish with a dollop of sour cream in the center.