Preheat a grill or grill pan to medium high. Mix the turkey, jerk seasoning, apple, ¼ cup green onions and the panko in a bowl; season with salt and pepper. Form into four 1-inch-thick patties and make a small indentation in the middle of each with your thumb to prevent it from puffing up on the grill. Refrigerate.
Whisk the mayonnaise and chutney together in a large bowl. Add the cabbage, carrot and the remaining ¼ cup green onions, season with salt and pepper and toss to coat.
Brush the grill with oil or spray with nonstick grilling spray. Grill the turkey patties until browned and cooked through, 4 to 5 minutes per side.
Brush the cut sides of the buns with mayonnaise and sprinkle with jerk seasoning; toast on the grill, about 30 seconds. Serve the burgers on the buns and topped with slaw and sliced avocado.