Place all burger ingredients in a medium bowl. Use your hands to thoroughly combine without compacting the meat too much.
Divide the burger into 4 equal-sized portions. Remove one quarter and form into two small patties. Drizzle with olive oil. Set aside.
Divide the remaining mixture into 4 equal portions. Roll each portion into a ball and flatten with your hands to about 1/4 inch. Add 1 Tbsp sweet potato and 1 Tbsp caramelized onion in the center of two of the patties. Top each with a patty and pinch the ends together. Gently compress the stuffed burgers with the palm of your hand. Season with additional salt and pepper, if desired. Drizzle both sides with olive oil.
Grill for 3-5 minutes per side, or until the burgers reach your desired degree of doneness. Grill 2 apple slices about 1 minute on each side.
To serve, divide salad greens evenly between 2 plates. Top each with a stuffed burger and apple slice. Place the smaller burgers and raw apple slices on 2 additional plates for the kids!