Place chicken breasts flat sides down on a sheet of plastic wrap. Cover with another sheet of plastic wrap. Using a small saucepan or rolling pin, lightly pound breasts to about 1/3-inch thickness. Pat breasts dry with paper towel.
In a bowl, combine breadcrumbs, Parmesan cheese, and oregano. In another bowl, lightly beat egg and salt. Brush both sides of chicken breasts with egg, then lightly coat with crumb mixture.
Warm oil in a large, heavy skillet over medium-high heat. When oil shimmers, add chicken and cook about 3 minutes per side, until golden brown. Sprinkle mozzarella evenly on top of chicken breasts in last minute of cooking. Transfer breasts to serving plates.
Meanwhile, warm marinara sauce in microwave or small saucepan for 1-2 minutes. Spoon about 2 tablespoons sauce over each breast. Serve immediately.
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