Place 1 chicken breast between 2 sheets of wax paper. Pound with a meat mallet, rolling pin or heavy skillet until 1/4 inch thick. The thinner and more even, the better. Repeat with the remaining chicken.
Place the flour, eggs and breadcrumbs in separate shallow baking dishes and season each with salt and pepper.
Season the chicken on both sides with salt and pepper. Dredge each breast in the flour and tap off the excess. Next, dip into the egg wash and let any excess drip off. Finally, dredge in the breadcrumbs, pressing to help the crumbs adhere evenly on both sides. Transfer the breaded chicken to a wire rack set over a rimmed baking sheet. This step can be done up to 30 minutes in advance.
Heat 2 inches of the canola oil in a large high-sided saute pan over medium heat until the oil begins to shimmer. Cook 1 or 2 cutlets at a time in a single layer until golden brown on each side, about 3 minutes per side. Transfer the cutlets to another wire rack set over a rimmed baking sheet and immediately sprinkle with salt. Let the cutlets rest a few minutes while you make the salad.
Whisk together the lemon juice with some salt and pepper in a medium bowl. Slowly whisk in the 1/4 cup olive oil until the dressing is emulsified. Stir in the onion and let sit a few seconds. Taste and add more olive oil, if needed. Add the arugula and Parmesan and toss to coat; season with salt and pepper.
Place each cutlet on a plate and top with a large amount of salad. Serve lemon wedges on the side.