Cut the focaccia in half through the side to make two rounds (the top and bottom halves of the sandwich).
Spread the pesto over the top and bottom halves of the sandwich. Layer the arugula and ham over the pesto on the bottom half.
Coat a 10-inch nonstick skillet with cooking spray and warm it over medium heat (the skillet should be about the same diameter as the bread). Beat together the egg whites and pepper until blended, 1 minute, and add the mixture to the skillet. Cover, reduce the heat to medium low, and cook, without stirring, until the egg whites are just set, 4 to 6 minutes. Layer the cheese over the egg whites, cover, and cook until the cheese melts, 2 to 3 minutes.
Using a large spatula, slide the entire round of cooked egg and cheese onto the bottom half of the sandwich. Add the top half and press gently. Place the entire sandwich back in the pan, cover, and cook until warmed through, 2 minutes. Remove the entire sandwich to a cutting board and slice it as you would a pizza into six wedges.
Tip: If you can’t find sun-dried-tomato pesto (usually in the pasta aisle of the grocery store), basil pesto works just as well.