In a large mixing bowl, combine ground chicken, minced lemongrass, garlic, shallot, fish sauce, sugar, black pepper, and minced chili until well incorporated.
Form the mixture into 4 equal-sized patties, approximately 1-inch thick, and refrigerate for 15 minutes to set.
Prepare the ginger slaw by whisking together the grated ginger, lime juice, rice vinegar, sesame oil, and a pinch of sugar in a separate bowl.
Add shredded cabbage, carrots, and cilantro to the dressing; toss to coat and set aside at room temperature.
Preheat a grill or non-stick skillet over medium-high heat and lightly coat with oil.
Cook the chicken patties for 6 to 7 minutes per side until the internal temperature reaches 165 degrees Fahrenheit (74 degrees Celsius).
Lightly toast the buns until golden brown.
Assemble the burgers by spreading mayonnaise on the bottom bun, placing the chicken patty, topping with cucumber slices, and adding a generous portion of the ginger slaw before closing with the top bun.