Place chicken breasts on cutting board. Slice each chicken breast in half horizontally to create 4 equally-sized cutlets. If cutlets are uneven or are more than ½-inch thick, place cutlets between 2 sheets parchment paper and pound chicken until cutlets are roughly ¼-inch thick.
Liberally season chicken cutlets on all sides with salt and fresh pepper. Set cutlets aside.
Place flour in shallow bowl. Add salt and pepper to taste, then whisk ingredients together until fully incorporated. Set flour mixture aside.
Crack eggs directly into second shallow bowl. Beat eggs with whisk or fork until eggs are fully combined. Mixture should be pale in color and no longer streaky. Set egg mixture aside.
Add breadcrumbs and grated parmesan to third shallow bowl. Mix ingredients together until fully incorporated, then set breadcrumbs mixture aside.
One by one, place chicken cutlets in flour mixture, dredging both sides through flour completely. Shake any excess flour back into bowl, then transfer floured chicken to egg mixture.
Dip cutlet into egg mixture until submerged, then hold cutlet over bowl and allow excess egg to drip back into bowl.
Place cutlet in breadcrumb mixture and coat both sides completely, pressing cutlet into breadcrumbs lightly to ensure breadcrumbs stick.
Shake any excess breadcrumb mixture back into bowl, then transfer cutlet to large plate. Repeat until all cutlets have been completely breaded in flour, egg wash, and breadcrumbs. Set cutlets aside.
Line baking sheet with layers of paper towels. Place wire cooling rack over paper towels and set baking sheet aside.
Heat cast-iron skillet over medium-low heat. When pan is warm, add olive oil. Swirl pan to fully distribute oil, then continue heating skillet until oil is hot and shimmery.
When oil is ready, place cutlets in skillet, working in batches as needed to avoid overcrowding. Pan-fry cutlets 2 to 3 minutes or until golden brown, then flip cutlets over and fry another 2 to 3 minutes. When cutlets are golden brown on both sides and reach 160° Fahrenheit internally, transfer cutlets from skillet to wire cooling rack. Repeat with any remaining cutlets.
Allow chicken cutlets to rest on cooling rack 3 to 5 minutes. While chicken rests, plate fresh arugula and garnish with parmesan shavings, olive oil, and sprinkle of sea salt.
Once chicken has rested for desired length of time, divide chicken between serving plates with arugula salads. Garnish with additional lemon wedges and sea salt if desired and serve warm.