Preheat the oven to 425°F (220°C). Line a rimmed baking sheet with aluminum foil and place a wire rack on top. Lightly brush the rack with a small amount of the olive oil to prevent sticking.
Place one chicken breast between two sheets of plastic wrap or in a resealable bag. Using a meat mallet or rolling pin, pound each breast to an even thickness of about 1/4 to 1/2 inch. Repeat with all 4 breasts.
Season both sides of the pounded chicken breasts with the 1/2 teaspoon salt and the 1/2 teaspoon pepper.
In a small bowl, lightly beat the 2 large eggs.
In a large shallow bowl, combine the 1 cup seasoned panko bread crumbs, the 1 cup seasoning fine bread crumbs, and the 1/3 cup freshly grated parmesan cheese. Mix to combine.
Working one piece at a time, dip a chicken breast into the beaten eggs, allowing excess egg to drip off, then press it firmly into the breadcrumb mixture so the crumbs adhere evenly. Place the breaded breast on the prepared wire rack. Repeat with the remaining breasts.
Lightly brush or drizzle a small amount of olive oil over the top of each breaded chicken breast so the coating is lightly moistened (use the olive oil from the ingredient list).
Bake the chicken on the wire rack in the preheated oven for 10 to 12 minutes. Using tongs, gently flip each piece, brush or drizzle a little more olive oil over the top, and return to the oven for another 10 to 12 minutes, until the coating is golden and the internal temperature reaches 165°F (74°C).
While the chicken bakes, make the arugula salad: place the 6 to 8 cups arugula greens in a large bowl. Add a pinch of salt and a pinch of pepper.
Squeeze the juice of 1 to 2 lemons over the arugula and drizzle with 1 to 2 tablespoons olive oil. Toss the arugula until evenly coated. (Use the amount of lemon juice and oil you prefer within the listed ranges.)
Add the 1/4 cup freshly shaved parmesan to the dressed arugula and toss gently to combine.
When the chicken is done, transfer to a clean plate to rest for 2–3 minutes. Serve the crispy baked parmesan chicken immediately with the lemon arugula.