Prepare the chicken cutlets for grilling by slicing each breast in half lengthwise. Place the chicken cutlet halves on a cutting board. Place a piece of parchment paper or a plastic bag over the chicken and use a mallet to flatten each piece. The chicken breasts should be even in height so they cook evenly on the grill.
Next, make the marinade. Combine the olive oil, soy sauce or tamari, honey, balsamic vinegar, and dried garlic in a large measuring cup or bowl with a spout. Use a whisk to combine. Use a measuring cup to scoop out and reserve ½ cup of the marinade. Set the extra marinade aside for the carrot slaw.
Pour the rest of the marinade in a large zip-top bag. Use tongs to transfer the chicken to the bag. Seal the bag securely and let the chicken sit in the marinade at room temperature for at least 10 minutes.
While the chicken is marinating, make the carrot slaw. Place the shredded carrots in a medium bowl and pour over the reserved marinade. Use tongs to toss. Sprinkle the top of the slaw with the chopped scallions and peanuts, if desired.
Turn on the grill to medium heat. Once it is hot, use your tongs to place the chicken breasts onto the grill. Cover the top and let the chicken cook for 5 minutes. Turn once and cook for another 5 minutes. For safe eating, use a meat thermometer to ensure the thickest part if the chicken has reached 165 degrees Fahrenheit.
Remove the chicken from the grill to a plate and cover with foil to let the chicken rest for 5 minutes.
Serve the chicken hot with a side of the carrot slaw.