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Preheat oven to 350 degrees.
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Place each chicken breast half between two sheets of plastic wrap on a cutting board. Using the flat side of a meat mallet or the back of a skillet, pound chicken breasts until they are 3/8 inch thick.
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Remove chicken from plastic wrap and set on a large plate; season with salt and pepper. Place flour and matzo each in separate large, shallow dishes. Beat together eggs and Parmigiano-Reggiano in another large shallow dish. Coat each chicken breast half evenly with flour, shaking off any excess. Then dip in egg and cheese mixture, followed by matzo to coat.
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Heat oil in a deep large skillet over medium-high heat. When oil is very hot, reduce heat to medium and, working in batches, add chicken. Cook chicken, turning once, until golden brown, about 3 minutes per side. Transfer to a paper towel-lined plate. Repeat process with remaining chicken, allowing oil to reheat between batches.
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Transfer chicken to a rimmed baking sheet. Top each breast with 1/2 cup tomato sauce and cover with mozzarella. Transfer baking sheet to oven and cook until cheese is melted and sizzling, about 10 minutes.
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Meanwhile, preheat broiler. Transfer chicken to broiler and broil until cheese is bubbling, about 1 minute. Serve immediately.