Marinate the chicken breasts in the olive oil, balsamic vinegar and garlic in a medium bowl for 15 minutes.
Place the bread in a food processor and pulse until medium-size crumbs are formed, about 10 pulses. Season with salt and pepper. Transfer the breadcrumbs (or panko) to a shallow baking dish or rimmed plate and set aside.
Whisk the egg in a medium bowl and place the flour on medium plate. Dip a piece of the chicken in the flour, then the egg and then the breadcrumbs, lightly pressing down on the chicken and then turning it over to coat the other side. Set the breaded cutlet on a plate and repeat with the remaining pieces of chicken.
Heat the vegetable oil in a large frying pan over medium-high heat. Add the cutlets to the pan and cook until brown on each side and cooked through, about 8 minutes total. Use a spatula to transfer the chicken to a paper-towel-lined plate to blot before serving. Serve with some greens on the side - baby spinach, arugula and chopped endive are all good choices.