For the pork milanese: Preheat the oven to 400 degrees F. Bring a medium pot of water to a boil.
Toss the potatoes with some olive oil in a bowl and sprinkle with salt and pepper. Pour onto a baking sheet in an even layer and transfer to the oven. Roast until golden brown, stirring halfway through, 15 to 20 minutes. Remove from the oven and set aside until ready to use.
Meanwhile, season the boiling water with salt. Add the string beans and blanch until tender yet crispy, 2 to 3 minutes. Remove the beans from the boiling water and shock in a bowl of ice water to stop cooking. As soon they are cold to the touch, drain on a paper-towel-lined plate. Set aside until ready to use.
Set a heavy-bottomed, high-sided saute pan over medium-high heat and fill with 1 to 2 inches of oil. Heat the oil to 375 degrees F.
Lay out a large piece of plastic wrap on a cutting board. Set the pork chop on the plastic wrap and cover with another piece of plastic wrap. Using a meat mallet, evenly pound the chop to approximately 1/2- to 3/4-inch thick.
To a shallow dish, add the flour. To a second shallow dish, add the eggs and whisk to combine. To a third dish, add the breadcrumbs. Dredge the pounded pork chop in the flour on both sides and shake off any excess. Then coat the chop in egg, again shaking off any excess. Lastly, add the chop to the breadcrumbs, pressing the breadcrumbs into both sides of the chop.
Test the temperature of the frying oil with a piece of breadcrumb; if it bubbles all around and doesn't stop, the oil is ready. Carefully add the pork chop to the oil and fry until golden brown, 2 to 3 minutes. Drain on a paper-towel-lined baking sheet and season with salt. While the pork is frying, make dressing.
For the salad and dressing: Add the shallot, vinegar and mustard to a bowl. Whisk in the olive oil and season with salt and pepper to taste.
Add the arugula, tomatoes, bocconcini, roasted potatoes and blanched string beans to a large bowl. Add half of the dressing and toss to combine. Add more dressing as needed. Taste and adjust the seasoning.
Set the fried pork chop on a plate and top with the salad.