Cut the chicken breasts in half horizontally so you have 4 thinner pieces. Place them between two sheets of parchment paper and pound to an even 1/4-inch thickness. Blot the cutlets very dry with paper towels and season generously on both sides with salt and pepper.

Set up a breading station: in one shallow bowl, whisk together the flour, 1 teaspoon salt, and 1/2 teaspoon pepper. In a second bowl, beat the eggs with the water until smooth. In a third bowl, rub the panko, Romano cheese, and Meyer lemon zest together with your fingers until the zest is evenly distributed.

Coat each cutlet in the flour mixture, shaking off any excess. Dip it into the egg wash and let the extra drip back into the bowl. Press the cutlet firmly into the panko-Romano mixture, turning to cover both sides completely. Place the breaded cutlets on a wire rack and repeat with the rest. Let them rest for 5 minutes so the coating sets.

Heat the olive oil and 1 tablespoon butter in a 10-inch skillet over medium-high heat until the butter is shimmering. Add the cutlets in a single layer, working in batches if needed, and cook for 2 to 3 minutes per side until deep golden brown and cooked through. Move the cooked cutlets to a clean wire rack to keep them crisp.

Pour off any excess fat from the skillet, leaving the browned bits behind. Add the chicken stock, Meyer lemon juice, and the remaining 2 tablespoons butter. Scrape up the browned bits with a wooden spoon and let the sauce simmer for about 1 minute until it thickens slightly. Taste and adjust the seasoning, adding red pepper flakes if you like a little heat.

Spoon the sauce over the cutlets right before serving so the crust stays crunchy. Garnish with fresh parsley and lemon slices and serve immediately, without letting the cutlets sit in the sauce.