Whisk peanut butter, hot sauce, soy sauce and ginger together. Remove ⅓ of the mixture and set aside. Gently fold ground chicken into the remaining ⅔ of the peanut sauce.
Form chicken mixture into 4 burger patties and grill until cooked through thoroughly, about 6 to 7 minutes per side.
While the burgers are cooking, use a vegetable peeler to make long, thin strips out of the carrot and cucumber. Place in a small mixing bowl and add minced cilantro, lime juice and salt, then mix to combine.
Once the burgers are cooked, set them on a plate to rest for 5 minutes. Assemble burgers by spreading peanut sauce on one side of each bun, adding a chicken patty, and topping each one with the carrot and cucumber salad.