Prepare the Steak

1. Pat the steak dry with paper towels, then season both sides with salt equal to about 1% of the steak’s weight. For a 1 lb (454 g) steak, use about 0.16 oz (4.5 g) salt, which is roughly: 1 1/2 teaspoons of Diamond Crystal kosher salt, 1 teaspoon of Morton kosher salt, or 3/4 teaspoon of table salt.
2. Place the steak on a wire rack set over a rimmed baking sheet and refrigerate uncovered for at least 1 hour, preferably 2-4 hours. While the steak dry-brines, caramelize the onions and prep the remaining ingredients.

Caramelize the Onions

1. Trim off the top and root ends of the onions, then cut them in half. Peel them, remove the outer layer, and place each half cut-side down. Slice thinly.
2. Heat a 12-inch (30 cm) stainless-steel skillet over medium heat for 1-2 minutes. Add the oil and heat for another minute, then add the onions and salt.
3. Cook for about 3 minutes, stirring occasionally, until the onions start to soften. Reduce the heat to medium-low and continue cooking, stirring every few minutes, until the onions are very soft and deep golden brown.
4. If they start sticking or look like they might burn, add about 1 tablespoon (15 ml) water, scrape up the browned bits from the bottom of the pan, and keep cooking. Add a little more water as needed to prevent burning.
5. While the onions caramelize, make the mustard aioli and sauté the mushrooms in a separate skillet.

Sauté the Mushrooms

1. Slice the mushrooms 1/4 to 1/2 inch (0.6-1.3 cm) thick.
2. Heat a skillet over medium-high heat for 1-2 minutes. Add 2-3 tablespoons of oil and heat for another minute, then add the mushrooms.
3. Cook for about 15 minutes, stirring every few minutes, until browned and tender. Once they’re done, transfer them to a bowl and set aside.

Make the Mustard Aioli

1. Combine 1 cup mayonnaise, 2 tablespoons mustard, 1 tablespoon fresh lemon juice, and 2 garlic cloves, finely minced or grated, in a small bowl.
2. Season with salt to taste, then refrigerate until ready to use.

Cook the Steak

1. Take the steak and mustard aioli out of the fridge about 15 minutes before cooking.
2. Turn on the exhaust fan or open a window. Heat a skillet over medium-high heat for 3 minutes. Add 2 tablespoons of oil and heat for another minute. While the skillet heats, pat the steak dry again (the drier the surface, the better the crust).
3. If the steak has a fat cap, like a New York strip or picanha, hold it fat-side down with tongs and sear for 60 to 90 seconds, until the fat cap is well browned. Then carefully lay the steak flat in the pan and cook, flipping every 30 seconds.
4. After about 2 minutes, start checking the internal temperature. Insert the thermometer horizontally into the thickest part of the steak, avoiding the bone and large pockets of fat. Remove the steak from the pan when it reaches the pull temperature for your preferred doneness: 110°F (43°C) for rare, 120°F (49°C) for medium-rare, 130°F (54°C) for medium, 140°F (60°C) for medium-well, or 150°F (66°C) for well-done. See the complete steak temperature chart above the recipe card.
5. Transfer the steak to a cutting board and let it rest for 5 minutes.

Assemble the Steak Sandwiches

1. Slice the rested steak very thinly against the grain (see Notes below). Split the ciabatta rolls lengthwise and place them cut-side up on a foil-lined baking sheet.
2. Spread mustard aioli over the bottom halves, then layer on the sliced steak, caramelized onions, sautéed mushrooms, and cheese.
3. Leave the sandwiches open and place them in the oven, as close to the heating element as possible. Broil for 3-5 minutes, just until the cheese melts and the top halves are nicely toasted.
4. Remove from the oven. Spread more mustard aioli over the top halves, add the arugula, close the sandwiches, and serve right away.