To prepare cakes, place bread in a food processor; pulse 10 times or until coarse crumbs measure 1 cup (discard remaining breadcrumbs). Set breadcrumbs aside.
Place chicken in food processor; pulse until ground. Combine chicken, chives, 3 tablespoons mayonnaise, seasoning, 1/4 teaspoon salt, egg whites, and breadcrumbs in a medium bowl; mix well (mixture will be wet). Divide mixture into 8 equal portions, shaping each into a 1/2-inch-thick patty.
Heat oil in a large nonstick skillet over medium heat. Add patties; cook 7 minutes on each side or until done.
To prepare aioli, combine 2 tablespoons mayonnaise and remaining ingredients in a small bowl. Serve with cakes.