Slice each chicken breast crosswise into rounds. Cover with plastic wrap and pound flat to 1/4" thick. In a shallow dish, combine milk, vinegar, cayenne, garlic powder and salt. Add chicken, cover and chill for 30 minutes to an hour.
Combine peaches, cucumber, red pepper, onions, and jalapeño pepper in a large non-reactive bowl. Add apple cider vinegar, sugar and salt and stir well. Reserve basil leaves to stir in just before serving.
Heat oil in a deep, heavy pot or dutch oven over medium-high heat to about 250° F. Whisk together flour, cayenne pepper, salt and pepper in a shallow dish. Shake excess liquid from a piece of chicken, dredge in flour mixture, then move to a plate. Continue with all remaining chicken then coat again with milk mixture and dredge for a second coat. Fry in batches for 5-7 minutes until golden brown.
Toast rolls with olive oil under a broiler, if desired. Spread mayonnaise and sriracha on the bottom half of each roll then add a pile of arugula. Top each with 1-2 pieces of fried chicken, a generous mound of pickled vegetables and bun. Enjoy immediately.