To butterfly the chicken: Slice the chicken breast down the middle horizontally lengthways to into 2 thin breast pieces. Season the chicken on both sides with salt and pepper. Coat with flour then dip into the egg and then into the breadcrumbs to coat well.
Heat half the oil in a nonstick pan over medium heat. Cook chicken for 2-3 minutes per side, until golden browned and an internal temperature of 160 degrees is reached using a thermometer. Set aside on a paper towel lined dish to drain excess oil.
Whisk together the remaining oil and vinegar in a bowl and toss with arugula, onion, and tomatoes. Season with salt and pepper to taste.
Place the arugula over top of the chicken and serve.