Cut each filet horizontally (across the grain) into three equal pieces. Gently pound filets between sheets of plastic wrap to 1/4-inch thickness using a flat meat pounder or a rolling pin.
Whisk together lime juice, olive oil, and salt and pepper to taste, then toss with watercress, cilantro, and mint.
Pat 2 paillards dry and season with salt and pepper. Heat 3/4 tablespoon of vegetable oil in a 10-inch heavy skillet over high heat until just beginning to smoke, then sauté paillards, turning once, 1 minute for medium-rare. Transfer to a plate. Dry and season remaining paillards and sauté in same manner with remaining oil.
