Prepare marinade: In a bowl or zip-top plastic bag, combine all marinade ingredients. Add chicken, turning to ensure that breasts are completely coated with marinade. Cover bowl with plastic wrap or seal bag and refrigerate until needed.
Set your rice cooker to the "regular" setting. Place 1 tsp oil in rice cooker bowl. Add shallots, stirring to coat, then close or cover with lid. Cook for about 5 minutes, stirring occasionally, until shallots have softened. Add mushrooms and cook for 8 more minutes, covered and stirring occasionally, until mushrooms are soft and have released water. Stir in farro and cook, uncovered and stirring occasionally, for 3 minutes.
Stir in broth. Place chicken, discarding leftover marinade, on top of farro mixture. Close or cover with lid and reset rice cooker to "regular" setting. The program should take about 1 hour, depending on the rice cooker. Chicken is done when its internal temperature reaches 165°F.
To serve, place 3/4 cup farro-mushroom mixture on each plate and top with 1 chicken breast. Sprinkle with parsley, dividing evenly.
Serving Size: 3/4 c farro mixture and 5 oz chicken