In a small skillet, heat vegetable oil over medium. Season chicken with salt and pepper; cook until browned and opaque throughout, about 6 minutes per side.
Meanwhile, make slaw: In a medium bowl, whisk 3 teaspoons vinegar, sugar, and sesame oil. Add cabbage, scallions, carrot, and ginger; toss to combine. Set aside.
Make sauce: In a small bowl, stir together peanut butter, soy sauce, and remaining teaspoon vinegar; if necessary, thin with water.
Mound slaw on a serving plate. Thinly slice chicken crosswise on the diagonal. Place alongside slaw; spoon a little sauce on the plate, and serve remaining sauce on the side.