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. Arrange chicken 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.
In a shallow dish, add flour. In a second shallow dish, add eggs and beat lightly. In a third shallow dish, combine panko, grated Parmesan, parsley, lemon zest, 1/2 teaspoon salt, and 1/2 teaspoon pepper. Stir to combine.
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).
Line rimmed baking sheet or large plate with 3 layers of paper towels. In a large skillet over medium-high heat, heat vegetable oil until just smoking, about 400 degrees (this can take a few minutes). Add 2 cutlets and fry until deep golden brown and cooked through, about 3 to 5 minutes per side.
Drain cutlets well on paper towel-lined plate and transfer to prepared wire rack in oven to keep warm. Repeat with remaining 2 cutlets.