Begin by pounding out the chicken breast gently between 2 pieces of plastic wrap.
Prepare a breading station with flour, whisked eggs and milk, and panko crumbs. Season all with salt and pepper. Coat the breasts in flour, egg then panko. Lay the coated breasts out on a flat tray and refrigerate to let the crust dry out (about 10 minutes) this will ensure there is less moisture and less chance your coating will break when frying.
Shallow fry the paillard in hot oil for approximately 2 to 3 minutes per side depending on how thick the breasts are. Drain well, season with salt and set aside when done.
For the dressing:
Put the anchovies, egg yolks, garlic, lemon juice, and water into a blender and process for 30 seconds until the mixture is smooth. With the blender running, pour the olive oil in slowly for the dressing to emulsify. Stir in the Parmesan, season with salt and a couple of grinds of black pepper; set aside. (Refrigerate the dressing if you will not be using it right away.)
Prepare the salad by splitting the grape tomatoes in half through the middle lengthwise, mixing all ingredients in a large mixing bowl. Add the dressing and toss the salad well.
To serve lay the warm, crispy chicken paillard on the base of the plate and top with the salad. Garnish with shaved parmesan cheese, a wedge of lemon and a drizzle of olive oil.
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs
Due to the slight risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell.