Preheat the oven to 425 degrees. Working one breast at a time, place chicken breasts on a cutting board, flat side down. Press the chicken with the palm of one hand and cut horizontally to make 2 cutlets. With the flat side of a meat pounder, pound the chicken between pieces of plastic wrap until about 1/4 inch thick.

Put the flour, beaten eggs, and bread crumbs in separate shallow dishes and season each with salt and pepper. Season each chicken cutlet on both sides with salt and pepper. Dredge each cutlet in the flour and tap off any excess. Dip in the eggs and let the excess drip off. Dredge in the bread crumbs, pressing the bread crumbs to adhere to the chicken. Transfer the breaded chicken to a baking sheet or large plate.

Place on a wire rack over a rimmed baking sheet. Heat 1/4 cup of the canola oil in a large saute pan over high heat until it begins to shimmer. Add 2 pieces of the chicken to the hot oil and cook until golden brown on both sides, about 3 minutes per side. Transfer the chicken to a wire rack on the baking sheet. Heat the remaining 1/4 cup canola oil and cook the remaining chicken.

Divide the mozzarella evenly among the chicken breasts. Place the baking sheet in the oven and bake until the cheese is melted and golden brown, about 5 minutes.

Remove the chicken from the oven. Put one chicken breast on each of the four plates, ladle the hot marinara sauce over the top, and sprinkle with grated Parmigiano Reggiano.

In a large bowl, whisk together the lemon juice and olive oil. Add the arugula and season with salt and pepper. Top each plate with some arugula and serve.