In a large shallow bowl or resealable plastic bag, whisk together balsamic vinegar, Worcestershire sauce, and 3 tablespoons olive oil to combine. Pat steaks dry with paper towel, then add to bowl or bag with marinade and turn to coat. Set aside to marinate at room temperature for about 10 minutes.

While steaks marinate, rinse all produce. Cut cucumber into .25-inch slices and place in a medium bowl. Add apple cider vinegar and 1/4 teaspoon salt, and toss to combine. Set aside to pickle until ready to serve. Cut tomato into .25-inch slices, place on a plate, and season with 1/4 teaspoon salt. Roughly tear basil leaves, discarding stems. Halve lemon.

Heat 1 tablespoons olive oil in a large pan over medium-high heat. Remove steaks from marinade, allowing excess to drip off, then season all over with 1/2 teaspoon salt and pepper as desired. When oil is shimmering, add steaks and sear, working in batches if needed, until browned and medium rare, 4-6 minutes per side. Transfer steaks to a plate and set aside to rest for about 5 minutes.

While steaks rest, in a medium bowl, whisk together horseradish, mayonnaise, and juice of 1/2 lemon. Season with 1/8 teaspoon salt and a generous amount of pepper.

Wipe pan from steaks clean and place over medium heat. Halve baguettes lengthwise. Carefully hover your hand over the pan—when you can easily feel the heat, add the baguette halves cut-side down. Toast, working in batches if needed, until golden, 4-5 minutes. Remove and slather toasted sides with horseradish sauce.

Drain cucumbers and return to bowl. Add basil and 1 tablespoon olive oil, tossing to coat. Slice steaks against the grain as thinly as possible (see Recipe Tip). Divide steak, tomatoes, and arugula evenly between baguettes then halve sandwiches on a diagonal. Serve sandwiches with pickled cucumber.