For the pesto: Combine the basil, 1/2 cup olive oil, the cheese, pine nuts and garlic in a food processor. Blend while drizzling in the remaining 1/2 cup oil to achieve the desired consistency. (The pesto should be fluid enough to squeeze from a squeeze bottle, if desired.) Season with salt and pepper.
For the sandwich: Using a sharp knife, butterfly the chicken breast starting from the tenderloin side. Spread open on a cutting board. Cover the chicken with plastic wrap and use a meat mallet or heavy frying pan to pound it to about 1/2-inch thick.
Crack the eggs into a shallow dish large enough to hold the chicken breast. Add a splash of olive oil and the cheese and whisk vigorously.
Put the breadcrumbs in a separate shallow dish.
Heat a heavy skillet over medium heat. Add 1/4-inch olive oil. Dip the chicken in the egg mixture, letting any excess drip off, then press the chicken into the breadcrumbs to coat both sides. Pan-fry the chicken until cooked through, about 4 minutes per side. Drain on paper towels or cardboard.
Toast the ciabatta rolls. Meanwhile, place the chicken on a griddle or in an oiled skillet, top with the prosciutto and provolone and cover with a metal bowl or lid to melt the cheese.
Spread the pesto on the bottom half of each roll. Cut the chicken in half and place on the rolls. Top with the tomato slices and arugula. Whisk the vinegar with a little bit of olive oil and sprinkle over the arugula. Cover with the top halves of the rolls.
This recipe was provided by a professional chef or restaurant and may have been scaled down from a bulk recipe. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.