Preheat an outdoor grill or grill pan to high. If using a grill pan, also preheat an oven to 400ºF.
Season the steaks with salt and freshly ground black pepper and drizzle with EVOO. If using an outdoor grill, cook the steaks until medium-rare, turning once, about 4-5 minutes per side. If using a grill pan, cook the steaks long enough to get some nice grill marks on each side, about 2-3 minutes per side. Transfer to a baking sheet and finish cooking the steaks in the oven until they are medium-rare, about 6-8 minutes.
While the steaks are cooking, combine the hearts of palm, cucumber, red bell pepper, red onion, romaine heart, lemon juice and 2 tablespoons of EVOO. Season with salt and freshly ground black pepper and toss to combine.