Preheat oven to 425 degrees. On a rimmed baking sheet, toss breadcrumbs and oil until well combined; spread on sheet. Cook, tossing once, until golden brown, 6 to 8 minutes. Transfer to a medium bowl. Place flour and eggs in separate bowls; season with salt and pepper. Place a rack on another rimmed baking sheet.
One at a time, place chicken breasts between two large pieces of plastic wrap. Using a meat mallet or bottom of a small heavy pan, pound until 1/2 inch thick. Coat chicken on both sides with flour (shaking off excess), then dip in eggs (allowing excess to drip off); dredge in breadcrumbs, pressing firmly to adhere.
Bake chicken on rack, without turning, until opaque throughout, 10 to 15 minutes.
In a bowl, whisk together remaining 2 tablespoons oil and lemon juice; season with salt and pepper. Add arugula and onion; toss. Top chicken with salad; serve with lemon wedges on the side.