Put the flank steak in a glass baking dish and pierce all over with a fork. Pour the pickling liquid over it; turn over several times to coat, and let stand, turning frequently, for 15 minutes. Preheat a stovetop grill pan over medium-high heat. Drain excess liquid from the steak, season well with salt and pepper, and grill, turning once, until medium rare, 3 to 4 minutes per side. Transfer to a cutting board and let stand at least 10 minutes before cutting against the grain into thin slices.
In a medium bowl, mix together the farmers cheese and chopped cherry peppers. Season with salt and pepper; to taste and add a few drops of water to make the cheese moist and spreadable. Lay a lavash sheet lengthwise on a work surface and spread 1/4 of the cheese mixture over it, leaving a 1-inch border. Evenly scatter 1/2 cup arugula leaves, followed by 1/4 of the sliced steak over the cheese. Fold the ends in toward the center, then, roll the lavash tightly from the bottom into a roll. Repeat with remaining ingredients. Wrap the rolls tightly in a sheet of plastic wrap or parchment paper and refrigerate until ready to serve.
To serve, slice the roll-ups into 1-inch segments and arrange them on a serving platter.
Note: Regular burrito-sized tortillas or flavored wraps can be used in place of lavash.