Whisk all the balsamic vinaigrette ingredients together in a small bowl. Place the chicken in a bowl and pour half the vinaigrette over the chicken, wrap in plastic wrap and place in the fridge for about 30 minutes.
Chop the vegetables into ½-inch pieces and toss with ½ tablespoon olive oil in a large bowl.
Fire up the grill and grill the vegetables until they are soft and tender. Grill the chicken until it registers 165 °F on an internal thermometer.
Return the vegetables to the same large bowl and stir in the can of chickpeas (be sure to rinse and drain the chickpeas first!)
In a medium saucepan, bring the 1½ cups of water to a boil over high heat, then add the 1 cup of uncooked bulgur and ¼ teaspoon salt. Turn the heat all the way down to low and cook, covered, for about 10 minutes until the bulgur is tender and all the water has been absorbed.
Cut the chicken into bite-sized pieces. Toss together the bulgur the vegetables, the chicken, and the remaining half of the balsamic vinaigrette.
Sprinkle with the chopped fresh basil