To cook the chicken, in a pan over high heat, add 1 tablespoon of the cooking oil, then add the ground chicken and stir-fry until fully cooked and lightly browned, then add the garlic and stir-fry briefly until fragrant.
To cook the vegetables, in the same pan over high heat, add the remaining 1 tablespoon of cooking oil, then add the carrot and cabbage and stir-fry for about 1 minute until slightly softened.
To finish the filling, return the chicken to the pan, then add the scallions, oyster sauce, hoisin sauce, and white pepper and mix well, then remove from heat and let cool slightly.
To soften the rice paper, in a bowl, whisk the egg, then brush a thin, even layer over both sides of the rice paper and let it sit for a few seconds until just softened.
To wrap the dumplings, place a small amount of filling near the bottom, then roll tightly and create layers as you wrap to form a compact dumpling.
To pan-fry the dumplings, in a pan over medium heat, add the cooking oil, then place the dumplings in a single layer with space between each one and cook for about 2 minutes per side, turning as needed, until golden and crispy.
To serve, transfer to a wire rack so they stay crisp, then serve hot with soy sauce, vinegar, and chili oil.