For the arugula salad: Whisk together the vinegar, mustard and some salt and pepper in a large bowl. Slowly whisk in the oil until emulsified. Add the arugula and toss to coat the leaves with the vinaigrette. Season with salt and pepper.
For the chicken: Place the flour, eggs and breadcrumbs in 3 separate shallow bowls and sprinkle each with salt and pepper.
Place each chicken breast between 2 pieces of plastic wrap and pound to 1/4-inch thickness.
Sprinkle the chicken on both sides with salt and pepper, dredge each breast in the flour and tap off excess. Dip into the egg wash and let any excess drip off, and then dredge on both sides in the breadcrumbs. Place on a baking rack set over a baking sheet.
Preheat 2 large nonstick saute pans over high heat. Add 3 tablespoons of the butter and 1 tablespoon of the oil to each pan and heat until the butter is melted and sizzling. Place 2 breasts in each pan and cook until golden brown, about 4 minutes. Flip the breasts over and continue cooking until golden brown and the chicken is cooked through, about 3 minutes longer. Remove each breast to a large plate and immediately top each breast with a few slices of cheese, a few slices of ham and some of the arugula salad.