To make the onions, slice the onions in half through the stem. With a mandoline or vegetable slicer or a thin sharp knife, cut lengthwise into 1/8-inch slices. (You should have about 6 cups.)
In a large nonstick skillet, melt the butter over moderately low heat. Add the onions, sprinkle with 1/2 teaspoon of the salt, and toss to coat. Cover and cook until the onions have released their liquid, about 13 minutes.
Uncover the pan, increase the heat to moderate, and cook slowly, stirring occasionally, until the liquid has evaporated, about 10 minutes. Sprinkle the onions with the sugar and continue cooking, stirring frequently, until they are golden brown and caramelized, about 10 minutes longer. Sprinkle with the balsamic vinegar and toss until the vinegar has evaporated. Add the remaining 1/4 teaspoon salt, and season generously with pepper. Remove from the heat and cover to keep warm.
Meanwhile, prepare the pepper rub: In a small heavy skillet, toast the peppercorns and allspice over moderate heat, shaking the pan occasionally, until fragrant, about 3 minutes. Transfer to a blender or spice grinder and grind to a fine powder. Strain the spices into a small bowl and return the coarse bits to the blender. Blend again and strain.
Pat the steaks dry with paper towels; rub lightly with a little of the oil. Sprinkle lightly with salt and massage the ginger into the steak; then rub the pepper rub over them.
Heat a grill pan or heavy nonstick skillet over high heat. Lightly oil the grill pan, if using, or swirl the remaining oil in a nonstick skillet. Add the steaks to the pan and cook until little droplets of blood form on the surface, about 4 minutes. Turn the steaks over and continue cooking until a droplet or so of blood forms on the top again, another 3 to 4 minutes, for rare. Transfer the steak to a cutting board and let rest for 5 to 10 minutes.
With a thin sharp knife, slice the steaks on a slight angle against the grain. Sprinkle the meat with a little salt and arrange the slices warm diner plates. Nestle a mound of onions next to the steak.
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