Place the chicken cutlets between 2 pieces of plastic wrap, parchment paper, or in a gallon freezer bag, in a single layer.
Using a meat mallet or rolling pin, pound the chicken to an even thickness, about ¼-inch thick, being careful not to smash the chicken.
Pat the chicken dry with paper towels and season both sides with the garlic powder, paprika, salt, and pepper.
Heat 2 tablespoons of the olive oil in a large skillet over medium-high heat.
Working in batches to prevent crowding the pan, add the chicken and cook for 2 to 3 minutes per side, until golden brown on both sides.
Transfer the chicken to a plate and cover with aluminum foil to keep warm.
Return the pan to the heat, reduce the heat to medium-low, and add the butter to the pan.
If the butter starts to smoke, remove the pan from the heat until the smoke disappears.
Cook the butter over medium-low heat until it turns golden brown and smells nutty, scraping up any browned bits from the bottom of the pan.
Add the garlic and cook for 30 seconds, being careful not to burn the garlic.
Add 2 tablespoons of the lemon juice and ½ teaspoon lemon zest.
Return the chicken to the pan and turn to coat with the butter sauce (sauce will be thin).
Cook for 2 to 3 minutes, until the chicken reaches an internal temperature of 165 degrees F.
Meanwhile, transfer the arugula to a large bowl, add the remaining tablespoon of olive oil and tablespoon of lemon juice and toss to coat.
Season the arugula with salt and pepper.
Transfer the chicken to a plate and spoon the lemon butter sauce over top.
Arrange the arugula alongside the chicken (or on top of the cutlets) and serve.