Preheat oven to 375 F. Fit a wire cooling rack over a baking sheet and mist with nonstick cooking spray; set aside.
In one of two shallow bowls, whisk together the egg and 1 tablespoon water. In the second bowl, combine the panko, breadcrumbs, Parmesan, salt, pepper, and garlic powder. Working one at a time, dredge the chicken in the egg mixture and then in the breadcrumb mixture, coating both sides completely.
Heat the olive oil in a large nonstick skillet over medium-high heat. Once hot, add the chicken and cook 2-3 minutes per side of until golden brown; transfer to the baking sheet. Bake for 20-25 minutes.
Meanwhile, prepare the arugula salad. Whisk together the shallot, Dijon, vinegar, olive oil, sugar, honey, salt and pepper. Drizzle the dressing over the arugula and toss to combine.
With 5 minutes left on the chicken cook time, wipe the skillet clean of any crumbs and return to medium-high heat. Cook the 3 eggs to your liking - over easy, medium, or hard.
To assemble, plate each chicken breast and top with 1 egg and ⅓ of the salad mixture. Serve immediately.