Heat the oil. Add a tablespoon of olive oil (or avocado oil) to a large frying pan or wok over high heat.
Cook the chicken. Once the oil is hot, add the chicken and season with salt and white pepper. Cook for about 3 minutes per side, until golden brown. Remove the chicken from the pan and set it aside, keeping the juices.
Sauté the aromatics. Return the pan to the heat, then add the garlic and ginger. Add a drizzle of cooking oil if needed. Sauté for about 30 seconds, until fragrant.
Add the veggies. Toss in the cabbage and carrots, stirring frequently. Cook for about 2 minutes until they're tender-crisp.
Make the sauce. In a small bowl, whisk together chicken broth, soy sauce, maple syrup, lime juice, and sesame oil.
Combine everything. Return the cooked chicken to the pan and pour in the sauce. Stir everything together and cook for another 2 minutes.
Serve and garnish. Remove from heat, top with green onions and sesame seeds, and serve immediately.