Cut each chicken breast half in half lengthwise. Season with salt and pepper. Place the chicken in the slow cooker (crockpot).
Heat canola oil in a large skillet set over medium heat. Add ginger and cook, stirring, for 2 minutes. Add garlic and cook for additional 30 seconds. Remove from heat.
Add ginger ale, chicken broth, hoisin sauce, and soya sauce. Stir with a whisk until the hoisin sauce dissolves. Pour the liquid over the chicken.
Set the slow cooker at high heat and cook for 1 hour. Turn to low heat and cook until chicken is just cooked through, 1½ to 2 hours (mine took 1½ hours).
Remove the chicken to a cutting board, let cool slightly and shred with 2 forks or your fingers.
Transfer the liquid to a medium-sized saucepan set over medium heat and boil until the sauce reduces by half. In a small bowl, stir together arrowroot or cornstarch and water. Whisk into the sacue and cook for an additional 5 minutes.
Transfer the shredded chicken and sauce back to the slow cooker and stir to combine. Set at low to keep warm.
The Slaw:
In a small bowl, whisk together rice vinegar, sugar, ginger, sesame oil, canola oil, lime zest, lime juice, and jalapeno pepper. Set aside.
Separate the cabbage leaves into a large bowl. Add carrots, scallions, and cilantro, and toss well.
Pour the dressing into the cabbage mixture and toss again. Let stand for 10 minutes, tossing occasionally.
Putting It Yogether:
Toast the hamburger bun halves until golden brown.
Divide the hamburger bun halves across several plates and top each with shredded chicken and the slaw. Serve any extra slaw on the side, if desired.
Notes
Please note that I am not a medical or nutritional professional. I provide nutritional information for my recipes as a courtesy to my readers. It is calculated using the built-in recipe card calculator. While I attempt to provide information that is as accurate as possible, you should calculate the nutritional information independently before relying on it.