Preheat an outdoor grill to high. (If you hold your hand about 3 inches from the grill grate, you should have to move it within 3 seconds.)
Arrange a chicken breast between two sheets of plastic wrap. Using a meat mallet or small heavy pan, pound the chicken evenly until it is 1/4 inch thick. Remove the plastic wrap and repeat with the remaining chicken breasts; put the paillards on a baking sheet and brush both sides of each piece lightly with oil. In a small bowl, combine 1 teaspoon salt, 1/2 teaspoon pepper and the garlic powder and sprinkle it evenly over both sides of the chicken pieces.
Lightly oil the grill grate with an oil-soaked towel. Grill the chicken, turning once, until cooked through and grill marks appear, about 2 minutes per side. Transfer to a clean platter.
In a large bowl, whisk the lemon zest and juice, mustard and 2 tablespoons oil until combined; season with salt and pepper to taste. Add the arugula to the bowl and toss to evenly coat.
Put a piece of chicken on each of four plates and top with a mound of arugula. Garnish with cheese and more pepper to taste and serve immediately.