In a small bowl, whisk the balsamic, olive oil, and garlic. Pour into a large, resealable plastic bag and add the chicken. Press the air from the bag, seal, and refrigerate for at least 30 minutes.
In a small bowl, whisk the apple cider vinegar, garlic, Dijon, honey, olive oil, salt, and pepper. Let stand while you assemble the rest of the salad.
Heat your gas grill or indoor grill pan over medium-high heat. Remove the chicken from the marinade, discard the remaining liquid, and grill the chicken until browned and cooked through, 12 to 15 minutes, turning halfway. Transfer the chicken to a cutting board and slice.
In a large bowl, toss the arugula with the radish slices and red cabbage. Add the cheddar, almonds, dried apples, and the vinaigrette and toss to coat. Divide the salad among your serving plates or bowls and top with avocado slices and chicken. Serve immediately.