3 romaine hearts (1 pound), cut crosswise into 1-inch strips
3 cups baby arugula
12 cherry tomatoes, halved
Preheat the oven to 375°. In a food processor, blend the mayonnaise with the lemon juice, mustard, anchovies, garlic, Worcestershire, Tabasco, 1/4 cup of the Parmigiano-Reggiano and 2 tablespoons of the olive oil until smooth. Season the dressing with salt and pepper. On a large plate, spread 1/4 cup of the dressing all over the steak and let stand for 30 minutes.
Meanwhile, on a rimmed baking sheet, toss the rye bread cubes with the remaining 2 teaspoons of olive oil and season with salt and pepper. Toast in the oven for 10 minutes, turning once halfway through, until the croutons are golden and crisp. Let cool.
Light a grill. Season the steak with salt and pepper. Grill over moderate heat, turning once, until the steak is medium rare, about 14 minutes. Transfer the steak to a cutting board and let rest for 5 minutes.
In a large bowl, toss the romaine with the arugula, cherry tomatoes, rye croutons, the remaining 2 tablespoons of Parmigiano-Reggiano and the reserved dressing. Thinly slice the steak on the diagonal across the grain. Mound the salad on plates, top with the steak slices and serve.