Heat the oil in a large skillet over medium-high heat. When hot, add the celery, green onions, and garlic. Sauté until just browned, about 5 minutes. Turn off the heat and set aside to cool.
In a large bowl, combine the grated apple, turkey and remaining burger ingredients, as well and the cooled vegetables.
Stir everything together and form into four patties, each about 1/2 inches (4 cm) thick. Let them rest at room temperature for about 15 minutes, or keep them covered in the refrigerator until ready to grill.
While the meat rests, prepare the aioli. In a medium bowl, stir together all of the ingredients. Adjust the seasoning with salt and pepper.
Heat the grill to medium-high. Cook the burgers until interior reaches 175°F (80°C), about 5 minutes per side. Put a slice of cheese on each patty during the last minute of cooking time.
Transfer the patties to a clean plate and tent loosely with foil.
Grill or toast the buns and spread each side with a bit of the aioli, and build the burgers with bacon, lettuce, tomato, onion and cheese.
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