Pound the chicken. If you are starting with cutlets, skip this step. Slice each chicken breast horizontally to make four thin cutlets. Place in a single layer on a cutting board. Cover with plastic wrap or parchment paper and gently pound to an even 1/4-inch thickness. The most important part of this process is getting the breast pieces an even thickness. Don't pound so hard that they start breaking apart.
Season the chicken. Season both sides of the chicken with salt and freshly ground black pepper.
Prepare the breading. Place the 1/2 cup flour in one shallow dish. Whisk the 2 eggs and tablespoon of water in a second dish. Combine the 2 cups panko and zest of 1 lime in a third.
Bread the cutlets. Coat each cutlet in flour and shake off the excess. Dip into the egg mixture, then press firmly into the panko.
Fry until golden. Heat enough oil to cover the bottom of a large frying pan by at least 1/2 inch over medium heat. Test the oil to see if it is ready by dropping a pinch of panko in the oil. If it bubbles right away it is ready. If it takes a couple of seconds, give the oil another minute to heat up before frying.
Once oil is hot enough, fry one or two cutlets at a time (don't crowd the pan) for 2 to 3 minutes per side, until golden brown and the center reaches 165°F. Transfer to a wire rack or paper towel-lined plate, season lightly with salt, and repeat adding more oil as needed.
Serve immediatley with lime wedges for squeezing.