For the sundried tomato pesto: Put the sundried tomatoes and their oil in a food processor along with the basil, Parmesan and salt. Process to combine.
For the chicken Parmesan: Season the chicken cutlets on both sides with the salt and pepper. Cut them in half crosswise. Using 3 wide, shallow bowls, place the flour in the first bowl, beaten eggs in the second, and combine the panko, Parmesan and oregano in the third.
Heat the oil in a large skillet over medium-high heat. Dredge the chicken pieces in the flour to coat lightly, shaking off the excess. Dip the chicken into the egg and then into the panko mixture, pressing gently to adhere the crust to the chicken. Add the chicken to the skillet and cook until golden brown on the first side, 3 to 4 minutes. Gently flip all of the chicken over and layer 1 mozzarella slice on top. Cook until the second side is golden brown and the mozzarella has melted, about 4 more minutes.
Transfer the chicken to a serving platter, mozzarella-sides up. Spread about 1 tablespoon of Sundried Tomato Pesto on top. Put a tomato slice on 8 of the pieces, and then top them with the remaining 8 pieces, mozzarella sides down, to make little sandwiches. Serve immediately.