Using a flat meat pounder, lightly pound each chicken breast half to an even thickness of about 1/2 inch. Cut each in 1/2 crosswise to make a total of 8 chicken breast pieces.
Season with 1 teaspoon Kosher salt and 1/2 teaspoon freshly ground black pepper.
Add eggs to another wide, shallow bowl. Beat well.
Dip a chicken breast in eggs.
Remove, allowing excess to drip off, and place in panko mixture. Turn and pat to adhere breading to chicken.
Place on a platter.
Repeat with all remaining chicken breasts.
Line a baking sheet with paper towels.
Pour 3 tablespoons oil into a medium skillet. Place over medium heat.
When oil is hot and shimmering, carefully add 4 chicken breasts. Cook until golden brown, turning after 3 minutes and adjusting the heat as needed so the chicken cooks steadily without over-browning, about 7 minutes total.
Remove to a paper towel-covered plate to drain.
Move cooked breasts to the second baking sheet place in preheated oven to keep warm while frying the remaining chicken.
Add remaining 3 tablespoons of oil and fry remaining chicken.
In a medium bowl, add arugula, tomatoes and lemon juice. Toss together. Season with salt and pepper, to taste.
Place 2 chicken pieces on each plate.
Top each with equal amounts of prepared salad.
Using a swivel vegetable peeler, shave as many Parmesan curls as you like on top.
Add lemon halves and serve immediately, hot.