Coat a stove-top grill pan or griddle with cooking spray and set over medium-high heat to preheat. Season both sides of chicken with salt and black pepper. Place chicken on hot pan and cook 3 to 5 minutes per side, until cooked through.
In a small bowl, whisk together rice wine, vegetable oil, wasabi paste, hot mustard, and ginger. Set aside.
Heat sesame oil in a large skillet over medium-high heat. Add slaw mix and soy sauce and cook 2 to 3 minutes, until wilted.
Arrange wonton wrappers on a flat surface. Top each wrapper with 1 heaping teaspoon of slaw mixture. Pull up sides and pinch the middle together, making a purse. Heat peanut oil in a large skillet over medium-high heat. Add purses and cook 1 minute, until bottom is golden brown. Add broth and bring to a simmer. Cover and steam 3 to 5 minutes, until purses are tender and translucent.
Serve 4 of the chicken breast halves with all of the vinaigrette spooned over top and all of the beggar's purses on the side. Reserve remaining chicken for stroganoff, if desired.