Slice each of the 4 medium chicken breasts in half horizontally to yield 8 thin cutlets; place each cutlet between sheets of plastic wrap or parchment and pound to about 1/4-inch thickness.
Season both sides of the cutlets evenly with 1 tsp of the fine sea salt and 1/2 tsp black pepper.
Set up a breading station with three shallow dishes: put the 1/2 cup flour in the first, whisk the 2 eggs with 1 Tbsp water in the second, and combine the panko, 1/2 cup Parmesan, 1/2 tsp garlic powder, 1 tsp Italian seasoning, and the remaining 1/2 tsp salt in the third.
Dredge each cutlet lightly in flour, shaking off excess, dip into the egg wash, then press into the breadcrumb mixture so the coating adheres; place breaded cutlets on a tray and repeat with remaining pieces.
Let the breaded cutlets rest 5–10 minutes to set the coating while you heat the oil.
Pour enough oil into a large skillet so it comes about halfway up the sides of the chicken and heat over medium until shimmering and hot (a breadcrumb should sizzle immediately); maintain oil temperature around 350–375°F.
Sauté the cutlets about 2 minutes per side, or until deep golden and the internal temperature reaches 165°F on an instant-read thermometer.
Transfer cooked cutlets to a wire rack or paper towels to drain, sprinkle with a pinch of coarse flaky salt, squeeze with lemon juice, and serve with extra lemon wedges and chopped parsley.