Preheat a grill or grill pan to medium high. Mix the turkey, jerk seasoning, apple, and half of the green onions 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 in a large bowl. Add the cabbage, carrots, and the remaining green onions, season with salt and pepper and toss to coat.
Brush the grill with canola oil. Grill the turkey patties until browned and cooked through, 4 to 5 minutes per side.
Serve the burgers and slaw on buns brushed with mayonnaise and sprinkled with jerk seasoning.