Place the panko in a pie dish. In another pie dish, beat the eggs to blend. Line a large cutting board with plastic wrap. Place 2 swordfish steaks on plastic wrap, about 2 inches apart. Place another sheet of plastic wrap over the steaks. Using the flat side of a meat mallet or a heavy rolling pin, gently flatten the swordfish steaks to 1/4-inch thick. Remove top plastic wrap and sprinkle the swordfish with salt and pepper on both sides. Set aside the swordfish steaks and flatten and season remaining 2 steaks, using the plastic wrap to help again. Dip each steak into the egg, then the panko, coating completely and patting to adhere.
Working in batches, melt 2 tablespoons of butter with 2 tablespoon of light olive oil in a heavy large frying pan over medium-high heat. Place 2 coated swordfish steaks in pan and cook until brown and crisp and just cooked through in the middle, about 1 1/2 minutes per side. Transfer to a cooling rack set on a baking sheet and lightly tent with foil to keep warm (do not seal with foil or the breading will become soggy). Repeat with the remaining butter, olive oil, and 2 swordfish steaks.
Whisk the lemon juice and extra-virgin olive oil in a large bowl to blend. Add the arugula and watercress, and toss lightly to coat. Season the salad, to taste, with salt and pepper. Transfer the swordfish steaks to plates. Top with the salad and serve.