In the bowl of an electric stand mixer, combine the water, yeast, and sugar. Stir briefly and let stand for 5 minutes or until the top surface has a thin, frothy layer (this indicates that the yeast is active). Add 4½ cups of the flour, the oil, and salt. Fit the mixer with the dough hook and mix on low speed for about 1 minute or until the dough begins to come together. Increase the speed to medium. Continue to mix until the dough is slightly sticky, smooth, and elastic, about 10 minutes. Form the dough into a ball and place in a lightly oiled bowl. Turn it over to coat all sides and tightly cover the bowl with plastic wrap. Allow the dough to rise in a warm place until it has doubled in size, 1½ to 2 hours.
In a small bowl whisk the dressing ingredients.
Trim most of the exterior fat from the steaks. Allow to stand at room temperature for 15 to 30 minutes before grilling.
Prepare the grill for direct cooking over high heat (450° to 550°F).
Punch down the dough in the bowl. Transfer to a lightly floured surface and cut into 4 equal pieces. Cut parchment paper into 10-inch squares and lightly oil each sheet of paper on one side. Roll each piece of dough flat into rounds 8 to 10 inches in diameter. Lay each dough round on an oiled sheet of parchment paper and lightly oil the top of each round. Stack the dough rounds between the sheets of parchment paper and set aside on a sheet pan.
Lightly brush or spray both sides of the steaks with oil and season them evenly with salt and pepper. Brush the cooking grates clean. Grill the steaks over direct high heat, with the lid closed as much as possible, until cooked to your desired doneness, 6 to 8 minutes for medium rare, turning once (if flare-ups occur, move the steaks temporarily over indirect high heat). Remove from the grill, cover, and keep warm.
Reduce the temperature of the grill to medium heat. Lay 2 dough rounds over direct medium heat, with the paper sides facing up. Grab one corner of the paper with tongs and peel it off. Grill until the rounds are golden and marked on the underside, 2 to 3 minutes, rotating them occasionally for even cooking.
Turn the crusts over and distribute one quarter of the cheese over each crust, leaving a ½-inch border around the edges. Continue grilling over direct medium heat until the crusts are crisp and the cheese is melted, about 2 minutes, rotating the crusts occasionally for even cooking. Transfer the grilled crusts to a work surface. Repeat with the remaining crusts.
Place the greens in a salad bowl, pour the dressing over the greens, and toss to combine.
Cut the steaks into thin slices, removing any pockets of fat, and distribute evenly over the crusts, then top with equal portions of the salad. Fold each piadini in half and eat it like a sandwich or, for easier eating or sharing, cut each piadini in half with a serrated knife after folding.