Lay the chicken out on a cutting board. Hold your knife parallel to the cutting board and slice the chicken breasts in half so one breast is now two cutlets. Lay the chicken between two sheets of plastic wrap and pound out with a meat mallet to 1/4-inch thickness. Sprinkle them on both sides with salt and pepper.
Set out three casserole dishes and add the flour to the first. Whisk together the mustard and eggs in the second. Mix together the Parmesan, lemon zest and some salt and pepper in the third. Dredge each cutlet first through the flour, then in the egg mixture and finally in the mixture. Put the chicken pieces on a clean baking sheet and refrigerate for 1 hour.
Preheat the oven to 250 degrees F.
Heat the oil in a large nonstick skillet. Once hot, add the cutlets, in batches, and cook for 3 minutes on each side. Remove to paper towels to drain and season again with salt. Put the chicken pieces on a clean baking sheet and place in the warm oven while you cook the second batch.
Serve the cutlets with avocado and lemon wedges.