High-heat cooking oil (such as safflower or grape seed), as needed
4 whole-grain or sprouted whole-grain buns
1 cup fresh cilantro leaves
In a small saucepan on medium-high, bring 1/2 cup water, vinegar and cane juice to a boil. Remove from heat and transfer to a heat-proof bowl; let cool for about 5 minutes. Add radishes, onions, cabbage, carrot and 1/2 tsp salt. Cover and refrigerate for at least 30 minutes or overnight.
Meanwhile, season chicken with remaining 1/4 tsp salt and pepper. Heat a grill to medium-high and lightly oil grate. Add chicken and grill, turning halfway, until lightly browned and no longer pink inside, 5 to 6 minutes per side. Transfer to a cutting board and cover loosely with foil to keep warm. Add buns to grill and toast lightly.
Cut chicken into thin slices and divide among buns. Using tongs, remove vegetables from vinegar mixture, shaking slightly over bowl to remove excess liquid. Divide vegetables evenly between bread and top with cilantro.