Tear romaine into pieces into a bowl. Halve cucumber lengthwise and seed. Thinly slice cucumber crosswise and add to romaine, along with the carrots and baby corn. Chill romaine mixture, covered. Mince shallot, jalapeno, and garlic (wear rubber gloves when handling jalapeno) and in a large bowl stir together with lime juice, soy sauce, 1 1/2 tablespoons oil, fish sauce, and sugar until sugar is dissolved.
Pat steak dry and season with salt. In an 8 to 10-inch heavy skillet heat remaining tablespoon oil over moderately high heat until hot but not smoking and saute steak 4 minutes on each side for medium-rare. Transfer steak to a cutting board and let stand 5 minutes.
While steak is standing, stack the mint leaves and thinly slice. Toss mint with romaine mixture and all but about 2 tablespoons dressing until combined well. Divide salad between 2 plates and slice steak. Toss steak with remaining dressing and arrange on salads.