Preheat a grill or a large grill pan over high heat. In a large roasting pan, place the steaks in a single layer and coat evenly with the rosemary, then season with salt and pepper and drizzle with EVOO.
In a medium pot of boiling salted water, simmer the beets until tender, 5 to 6 minutes, then drain. Place the beets on a baking sheet or large platter with the fennel, scallions and asparagus and drizzle liberally with EVOO; add the mushrooms and use a pastry brush to coat evenly with EVOO. Season with salt and pepper and grill until well marked (a couple of minutes for the beets, scallions and asparagus; a few minutes on each side for the fennel and mushrooms). Remove the vegetables from the grill and let cool. Thickly slice the mushrooms and halve the fennel slices.
Grill the steaks for 2 to 3 minutes on each side for medium-rare, then let rest for about 5 minutes before thinly slicing.
In a medium bowl, whisk together the preserves, vinegar and lemon juice, then slowly whisk in the 1/3 cup of EVOO in a thin stream. Toss the arugula and romaine with half of the dressing and season with salt and pepper. Top the salad with the grilled vegetables and drizzle the remaining dressing around the salad. Top with the ricotta salata crumbles. Serve the sliced steak alongside the salad.