Whisk together the tomato paste, rice wine vinegar, honey, and soy sauce until smooth.
Add in the diced vegetables and minced garlic. Add red pepper flakes, if desired.
Season the chicken breasts with salt and pepper and add to the slow cooker.
Set the slow cooker on low to cook for 5 hours.
30 minutes before serving, add the carrots and remove the chicken. Use two forks to pull it apart. Return to the slow cooker.
In a small dish, make a slurry with cornstarch and water. This will thicken the sauce.
Turn the heat up to high for 30 minutes.
Serve over rice with a garnish of sesame seeds and scallions if desired.