For the chicken: Combine the chicken, buttermilk, lemon and salt in a resealable plastic bag. Seal the bag and toss to coat. Place in the refrigerator and marinate for at least 6 hours or up to overnight.
For the crispy prosciutto: Preheat the oven to 350 degrees F.
Spray a rimmed baking sheet with olive oil cooking spray. Lay the prosciutto flat on the baking sheet and spray with a few more spritzes. Bake until lightly toasted and crispy, about 10 minutes. Allow to cool slightly on the baking sheet, then transfer to a rack to cool completely.
For the mayo: Mix together the mayonnaise and pesto in a small bowl. Refrigerate until ready to use.
For the sandwiches: Mix together the flour, breadcrumbs and salt in a shallow bowl. Heat a large skillet over medium-high heat. Add the grapeseed oil to cover the bottom of the skillet. Remove the chicken from the buttermilk, removing as much of the buttermilk as possible, and dredge in the flour mixture. Place immediately in the hot oil. Repeat with the remaining chicken. Fry on the first side until golden brown and there is a ring of cooked chicken around the edges, about 5 minutes. Gently flip the chicken and continue to cook on the second side until evenly golden brown and cooked through, an additional 4 minutes. Place a slice of mozzarella on top of each and cover the pan for an additional minute to melt the cheese. Alternatively, you can remove the chicken to a small baking tray, top with the mozzarella and place under the broiler until melted and bubbly, about 2 minutes. Repeat with the remaining chicken.
Spread the cut sides of the buns evenly with pesto mayonnaise. Place a small amount of arugula on the bottom halves and top with a cutlet. Break a slice of crispy prosciutto on top. Close the sandwiches and enjoy.