Combine first 9 ingredients in a large zip-top plastic bag. Add chicken to bag; seal and marinate in refrigerator at least 6 hours or up to 24 hours, turning bag occasionally.
Remove chicken from bag, reserving marinade. Pour marinade into a large skillet; stir in broth. Arrange chicken in a bamboo steamer coated with cooking spray; cover with steamer lid.
Place bamboo steamer in pan; bring marinade mixture to a boil. Steam the chicken 18 minutes or until a thermometer inserted in center of breasts registers 165°F Transfer chicken to a platter.
Remove steamer from pan. Strain marinade mixture through a fine sieve into a bowl; discard solids. Return marinade mixture to pan; bring to a boil over high heat. Cook until reduced to 1/2 cup; about 2 minutes. Serve sauce with chicken.