Preheat the oven to 250 degrees F and place a sheet pan in the oven.
Dry the fish on both sides with paper towels. Combine the flour, 2 teaspoons salt, and 1 teaspoon pepper on a dinner plate. In a low, shallow bowl, whisk the eggs with 1 tablespoon water. Put the bread crumbs on another plate. Dredge the fish on both sides in the seasoned flour and dust off the excess. Dip both sides into the egg mixture and, finally, dredge both sides in the bread crumbs.
Heat 1 tablespoon butter and 1 tablespoon olive oil in a large (12-inch) saute pan and cook 2 fillets at a time over medium to medium-high heat for 2 to 3 minutes on each side, until cooked through. Transfer the fish in one layer to the sheet pan in the oven. Add more butter and oil and cook the rest of the fillets, transferring them to the sheet pan as they're done. Toss the arugula with enough Lemon Vinaigrette to moisten. Place one fillet on each plate and pile the arugula salad on top. Heat the capers in the saute pan for 30 seconds and sprinkle over the fish and salad. Sprinkle with grated Parmesan, a squeeze of lemon juice, and the fleur de sel. Serve hot.
In a small bowl or measuring cup, whisk together the lemon juice, olive oil, 1 teaspoon salt, and 1/2 teaspoon pepper.
Making vinaigrettes in a measuring cup allows you to measure and whisk at the same time.