Marinate chicken with pesto, garlic, red pepper flakes, lime juice and 1/2 teaspoon salt at least 1 hour, or overnight for best results.
Mix oil, balsamic vinegar, honey and 1/4 tsp salt in a small bowl.
Heat a grill over medium-high, be sure grates are clean and well oiled to prevent sticking.
Brush oil on each side of the vegetables and sprinkle with salt and pepper
Put vegetables on 1 large grill tray or directly on grill, and cook, turning constantly until the vegetables are cooked and golden, about 6 to 10 minutes. Set aside on a dish.
Put the chicken on the grill and cook about 4 to 5 minutes on each side until grill marks appear and the chicken is cooked through. If you prefer to finish them off in the oven, I suggest a grill pan or cast iron skillet - cook until the internal temperature reaches 160 degrees.
Transfer the chicken to a platter with the vegetables and pour the balsamic dressing over everything and serve.