In a bowl, combine 2 oz onion, sugar, salt and vinegar. Cover and refrigerate for at least 30 minutes.
Make an aioli: In a bowl, combine mayonnaise, smashed garlic, lemon juice, parsley, 3/4 tsp salt and black pepper. Mix well and set aside.
For each sandwich, spread onion jam on the bottom half of a bun. Place arugula over jam, then turkey, then a slice of cheese, then pickled red onions. Spread aioli on the top half of each bun, then close the sandwich and serve.