In a food processer, place garlic, basil, salt, pepper, sugar and oil, process until smooth. Transfer to a bowl and add chicken tenderloins. Toss with the rub and marinate for 15 minutes.
In a blender, add basil, parsley, mayonnaise, Dijon mustard, sugar and black pepper, blend until smooth. Transfer to a ramekin and reserve.
In a bowl, toss the onions with the canola oil, salt, black pepper and brown sugar.
Place onions on the green egg and grill at 350 degrees until tender approximately 15 minutes or until charred and soft.
After the chicken has marinated for 15 minutes, place on the Green Egg at 350°F, the tenders will char rather rapidly due to the sugar in the rub. Cook on both sides for 4-5 minutes. Chicken is cooked when it reaches and internal temperature of 165°F. Transfer to a plate and assemble the sliders
Cut tender in half and place both pieces on the bottom bun, add 1 tablespoon Basil Mayo and top with caramelized onions. Serve.