Preheat the oven to 450°F (230°C).
Make the sauce: Heat the olive oil in a saucepan over medium heat. Add the onion and cook until soft, then stir in the garlic for 30 seconds. Add the tomato puree, sugar, salt and pepper, and let it simmer gently while you prepare the chicken.
Prepare the chicken: Pat the fillets dry with paper towels. Place each one between two sheets of plastic wrap and pound it to an even ½-inch thickness.
Coat in the egg mixture: In a shallow dish, whisk the eggs with the garlic, parsley, onion powder, salt and pepper. Add the chicken and turn to coat. If you have time, let it sit for 15 minutes (or cover and refrigerate overnight).
Bread the chicken: Mix the panko and Parmesan in a second shallow dish. Lift each fillet out of the egg mixture, let the excess drip off, then press it into the panko mixture to coat both sides.
Fry the chicken: Heat ½ cup of the oil in a 12-inch skillet over medium-high heat. When it shimmers, add 2 fillets and fry for 2 to 3 minutes, until golden. Flip, lower the heat to medium and cook for 3 more minutes, until golden. Move to a wire rack set over a sheet pan. Wipe out the skillet, add the remaining ½ cup of oil and fry the other 2 fillets.
Top and bake: Spoon the sauce over the cutlets and lay the mozzarella slices on top. Bake for 5 to 8 minutes, until the cheese is melted and the chicken reaches 165°F (74°C). For browned cheese, broil for the last minute or two.
Garnish with the fresh parsley and serve hot, with pasta if you like.