Make the cranberry aioli: Microwave cranberry sauce in a microwave-safe bowl for 30 seconds, until lightly melted. Allow to cool slightly. In a small bowl, whisk together mayonnaise, cranberry sauce, dijon mustard, salt, and pepper. Set in the refrigerator to chill.
In a medium sized mixing bowl, combine ground turkey, apples, onion powder, garlic powder, salt, and pepper. Mix until spices and apples are evenly distributed.
Separate into 4 equal parts. Form each into a flat hamburger patty.
To grill: Preheat grill to 350 degrees F. Place burgers on the grates. Cook, covered, for 5 minutes, or until burgers release easily from the grill grates. Flip, and cook another 5 to 8 minutes, or until the internal temperature reaches 165 degrees F. Add a slice of cheese on top of each burger in the last 1 to 2 minutes of cooking to lightly melt.
To bake: Preheat oven to 400 degrees F. Place burgers on a baking sheet sprayed with nonstick spray. Bake burgers for 13 minutes, flipping halfway through, or until the internal temperature reaches 165 degrees F. Add a slice of cheese on top of each burger in the last 1 to 2 minutes of cooking to lightly melt.
Serve apple turkey burgers on hamburger buns spread with cranberry aioli and then topped with kale apple slaw.