Cut each chicken breast horizontally almost completely through, leaving one side intact. Open up like a book. Cover with plastic wrap and pound to an even 1/4 inch thick.
Whisk eggs in a shallow dish; season with salt and pepper. Put flour in another dish, season with salt and pepper, and whisk to combine. Heat 1 inch safflower oil in a large straight-sided skillet over medium-high until a thermometer registers 375 degrees (a bit of flour should sizzle immediately when dropped in).
Working with one breast at a time, season chicken with salt and pepper, then dredge in flour mixture, tapping off excess. Dip in eggs, letting excess drip off, then coat again in flour mixture. Immediately place in hot oil and cook until golden brown, about 1 1/2 minutes a side. Remove with tongs and let drain on paper towels. Season with salt. Repeat with remaining chicken, adjusting heat as necessary to maintain oil temperature.
Toss together peaches, brine, bocconcini, and olive oil in a medium bowl. Season with salt and pepper. To serve, top cutlets with peach mixture and arugula, sprinkle with basil, and drizzle with more olive oil and more brine.