In three large, shallow bowls, add the flour, eggs, and panko bread crumbs.  Season each with salt and pepper and stir to combine.

For the chicken: you can pound the chicken breasts to an equal thickness (about 1/2-inch), if desired, but splitting the chicken into 4 thinner pieces means you really don’t have to.  Season each breast with salt and pepper. Run the chicken breasts through the flour, tapping off the excess.  Run through the egg wash, allowing the excess to drip off and coat with panko, patting the crumbs in to ensure entire chicken breast is coated.

Preheat the oven to 300°. Preheat a large nonstick sauté pan over medium-high heat.  Add 3 tablespoons of the butter and 1 tablespoon olive oil and allow to melt and heat.  Place two breasts at a time in the pan and cook for 3 to 4 minutes on each side, until golden brown.  Transfer to a baking sheet while you cook the remaining breasts.  Pour off the fat in the skillet and add the remaining 3 tablespoons butter and 1 tablespoon olive oil, allowing to heat again.  Cook the remaining two breasts in the same way.  When all of the chicken is cooked, top with the brie and reamining prosciutto slices and transfer to the oven for 1 to 2 minutes, until the cheese just starts to melt.  

Meanwhile, make the arugula salad.  Whisk together the dressing ingredients until emulsified. Add the arugula just before serving and toss to coat.  Top each chicken breast with the arugula salad and serve.