Preheat the oven to 375 degrees F.
For the chicken breastl: To pound the chicken breast, line a cutting board with plastic wrap and place a breast portion on top. Place another piece of plastic wrap over the chop making sure it's completely covered. Use a meat mallet to pound the chicken evenly, making sure the middle and the edges are the same size, about 1/2 inch in thickness. Remove plastic from the top and set the pounded chicken aside; repeat with the rest of the chicken breast.
Set up 3 shallow baking dishes, one with the flour, one with the beaten eggs, and the third with the breadcrumbs. Season the flour and breadcrumbs with salt and pepper. Mix the Parmesan and parsley into the breadcrumbs. Season each breast with salt and pepper on both sides. Dredge a breast in the flour. Shake off any excess flour and then dip it into the egg mixture and then into the breadcrumbs. Once the breast is in the breadcrumbs, use your hand to scoop the crumbs up and over the meat and press to make sure it adheres and evenly coats the breast. Repeat the breading procedure with the remaining pieces.
Heat a 12-inch skillet over medium-high heat and add 1/4 cup blended oil. When the oil is hot, add two of the breaded pieces to the pan. Cook for 3 minutes or until golden brown, then turn the breast gently with tongs and cook for another 3 to 5 minutes on the other side. Add 2 tablespoons of butter, and when the butter is almost brown, drizzle the juice of 1 lemon into the pan and over the crisp chicken. Remove the pieces and place them on a sheet pan. Drizzle the pan drippings over the breast. Repeat with the remaining 1/4 cup oil, remaining chicken breast, remaining 2 tablespoons butter and remaining lemon juice.
For the salad: In a medium bowl, combine the arugula or spring mix, radishes, olives, onion slices and tomatoes. Squeeze the lemon over the salad, add salt and pepper and toss. Drizzle with extra virgin olive oil and toss again.
To serve: Place a chicken breast on a serving plate and add some salad on top. Finish with some Parmesan shavings. Repeat with remaining chicken breast, salad and Parmesan and serve.