Preheat oven to 200 degrees. Line a rimmed baking sheet with foil for easy cleanup. Top with a wire rack coated with nonstick spray or olive oil.
Cover a cutting board completely with a piece of plastic wrap. Add chicken breasts. Working with one breast at a time, place your palm on top of the breast and slice in half horizontally so you have two cutlets. Repeat with the second breast.
Arrange all four cutlets in a single layer on the plastic wrap-covered cutting board. Cover with a second piece of plastic wrap. Pound chicken breasts to an even thickness, about 1/4-inch. Remove top layer of plastic wrap and season breasts with salt and pepper.
Working with one cutlet at a time, dredge in flour and shake off excess. Dip in eggs and tap gently on side of dish to remove excess. Add to panko mixture and press gently until crumbs adhere. Transfer to a large plate and repeat with remaining cutlets (cutlets can be wrapped in plastic and refrigerated for up to 2 hours).
Drain cutlets well on paper towel-lined plate and transfer to prepared wire rack in oven to keep warm. Repeat with remaining 2 cutlets.
Wipe out pan with dry paper towel. On medium heat, melt 1 tablespoon butter until foaming. Add lemon juice, white wine, and reserved lemon slices. Boil over high heat until reduced by half, about 3 to 5 minutes.
Off heat, swirl in remaining 2 tablespoons butter and capers. Season to taste with salt and pepper. Serve chicken with sauce, lemon slices, and fresh parsley.