Set up a standard breading procedure by putting the flour, 2 eggs beaten with 1 tablespoon water, and panko mixed with 1/4 cup grated Parmesan in separate wide flat containers. Season the pork, generously, with salt on both sides. Using 1 hand for wet ingredients and the other hand for dry ingredients, coat the meat in the flour, then the eggs, and then the bread crumbs, pressing the panko onto the pork to adhere. Set aside until ready to cook.
Line a baking sheet with a silpat or parchment paper. Spread a tablespoon of the grated parmesan into a thin circle about 3-inches in diameter to form a frico. Repeat 3 more times. Put the baking sheet into the oven and bake until the cheese is melted and lightly browned, about 10 minutes. Remove the pan from the oven and reduce the heat to 200 degrees F.
In a large saucepan over high heat, bring 2 to 3 inches of water to a boil. Add the white vinegar and reduce the heat to bring the water to a bare simmer. Carefully crack the remaining 4 eggs into the water and cook until the whites are set but the yolk is still soft, about 2 to 3 minutes. Remove the eggs with a slotted spoon and place them into a bowl of ice water to stop the cooking. Keep the water on the fire to reheat the eggs when you are ready to serve.
In a large skillet over medium heat add olive oil until it is about a 1/2-inch deep. Check the temperature of the oil by dipping the edge of the pork into the oil, if it does not sizzle and bubble STOP and WAIT! Check it again in another minute. If the oil is smoking it is too hot. When the oil is at the proper temperature gently add the meat. Do not crowd the pan or you will end up with greasy pork, it may be necessary to work in batches. When the pork is beautifully golden brown on each side and cooked through, remove from the pan and blot on paper towels. Sprinkle with salt and put them in a warm oven on a sheet pan.
In another large skillet over medium-high heat, pour in a thin layer of olive oil. Add the onions and a pinch of red pepper flakes, season with salt, and cook for 3 to 4 minutes. Add the mushrooms, season with salt, and cook until they are soft and wilted, about 3 to 4 minutes. If the pan is looking dry add another few drops of olive oil. Add the arugula, radicchio and red wine vinegar. Toss to combine and add more olive oil if needed. Season with salt, if needed.
When you are ready to serve, remove the poached eggs from the ice bath and put them back into the hot water. Arrange a piece of pork in the center of each serving plate and top it with some of the salad. Remove each egg from the hot water and pat dry. Put it on top of the salad and sprinkle with a few red pepper flakes. Carefully remove a frico from the baking sheet and prop it up next to the egg. Sprinkle some extra Parmesan all around and drizzle with some olive oil. You rock!