Working with 1 chicken cutlet at a time, place the meat between 2 sheets of parchment paper or plastic wrap. Using a meat mallet or rolling pin, pound each cutlet to ¼-inch thickness. Season the chicken cutlets on both sides with salt and pepper.
In a shallow bowl, beat the eggs. Combine the panko and Italian seasoning in a separate shallow bowl. Dip each cutlet into the egg, allowing the excess to drip off. Press each cutlet into the seasoned panko to coat evenly.
Heat 2 tablespoons of the vegetable oil in a large skillet over medium-high heat. Once the oil is glistening, add 2 cutlets and cook, undisturbed, until golden brown on the bottom, 3 to 4 minutes. Flip the cutlets and repeat until the other side is golden brown, another 3 to 4 minutes. Transfer the chicken to a paper towel lined plate.
Add the remaining 1 tablespoon of oil to the skillet and repeat with the remaining cutlets.
Let the chicken rest for 5 minutes, then slice crosswise. Sprinkle with the flaky salt and serve with a wedge of lemon on the side.