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Whisk the pineapple juice, soy sauce, lime juice, ginger, and garlic together in a large bowl.
Transfer half the marinade to a gallon-size zip-top bag and add the chicken breasts. Press to seal all the air out of the bag so the chicken is submerged in the marinade, and let sit for an hour at room temperature or 4-6 hours in the refrigerator.
Cut the bok choy in half lengthwise and rinse thoroughly. Slice the carrots lengthwise into wide, thin strips with a Japanese slicer or peeler.
Preheat a gas or charcoal grill for two-zone grilling (one side direct, high heat; one side indirect, medium heat).
Place the chicken over the hottest part of the grill and cook for 10 minutes, flipping once. Transfer to the indirect heat side of the grill and cook for 10-12 minutes more, flipping occasionally, until a thermometer inserted into the thickest part of the breast reads 165. Remove the chicken from the grill and let rest for 5 minutes.
While the chicken grills, place the remaining marinade in a saucepan and bring to a simmer. Cook for 10 minutes until slightly thickened. Whisk the cornstarch together with 2 teaspoons water and add to the marinade. Cook for 1 minute more.
While the chicken rests, toss the bok choy and carrots with a drizzle of olive oil and sprinkle of salt and pepper. Place the bok choy directly on the grill grates and the carrots in a grill basket and cook alongside the chicken just until the vegetables are lightly charred and tender, 2-3 minutes per side for the bok choy and 6-8 minutes for the carrots.
Drizzle the marinade sauce over the chicken and vegetables before serving.