In a large bowl, combine 1 quart water and the lemon juice. Rinse artichokes. Starting with lower, outer petals, snap off and discard leaves near bases, down to those that are half green and half yellow. Slice green tops off cones. Cut stems flush with bases and trim off any remaining dark green. Cut artichokes in half lengthwise and, if the center is fuzzy or prickly, scrape out. Drop artichokes in lemon water as trimmed. Lift artichokes out and slice paper-thin in food processor, with a hand guard on a box slicer or mandoline, or with a sharp knife. Return slices to lemon water.
Rinse beef and pat dry; trim off excess surface fat. Rub 1 tablespoon olive oil all over steak.
Set steak on a barbecue grill over a solid bed of medium coals or medium heat on a gas grill (you can hold your hand at grill level only 5 to 6 seconds); close lid on gas grill. Cook, turning once, until rare (red to pinkish red in the center and pale pink near the surface; cut to test or insert a thermometer in center of thickest part--it should register about 125°), 12 to 16 minutes total, or medium-rare (pinkish red in center, gray near surface, 135° in center), 16 to 20 minutes total. Let rest in a warm place for about 5 minutes.
Meanwhile, in a large bowl, mix 1 tablespoon each olive oil and balsamic vinegar. Add arugula and mix gently to coat. Arrange arugula on a large platter. To the bowl, add 1 more tablespoon each olive oil and balsamic vinegar. Drain artichoke slices well and add to bowl; mix gently. Add salt and pepper to taste. Spread artichokes and dressing evenly over arugula.
Cut steak crosswise into 1/4-inch-thick slices and arrange, slightly overlapping, on arugula. Sprinkle lightly with salt and pepper. With a vegetable peeler, shave curls of cheese onto steak. Drizzle with 2 tablespoons olive oil; offer more oil to add to taste.