Pat the steak dry with paper towels, then season generously on both sides with salt and freshly ground black pepper. Let the steak sit at room temperature for 1 hour. This allows the seasoning to absorb and helps the steak cook evenly.

Peel the potatoes and slice them into thin matchsticks using a mandoline (use caution). Place the potato sticks in a large bowl of cold water and stir in 1 teaspoon of salt. Let soak for 30 minutes to draw out excess starch, which helps them crisp. After soaking, drain the potatoes well and dry thoroughly with clean kitchen towels.

Heat at least 2 inches/5 cm of oil in a deep pot or heavy saucepan to 350°F (175°C). Working in batches, fry the potato sticks for 3 to 4 minutes until golden brown and crispy. Remove with a slotted spoon and drain on paper towels. While still hot, sprinkle lightly with salt.

To make the sauce, whisk together the mayonnaise, mustard, garlic, olive oil, lemon juice, salt, pepper, and finely chopped celery (including any tender leaves) in a mixing bowl until smooth and fully emulsified. Set aside.

Heat a cast-iron skillet or heavy frying pan over high heat. Add a thin layer of olive oil and sear the steak for 2 to 3 minutes per side, depending on thickness, until browned and medium-rare. In the final 2 minutes of cooking, add the onion quarters to the pan. Let them char slightly and soften, turning occasionally. Drizzle the onions with balsamic vinegar and continue cooking for another 2 to 3 minutes until caramelized. Transfer both steak and onions to a cutting board.

Let the steak rest for 5 minutes before slicing it thinly against the grain. Gently separate the onion layers.

To assemble, slice open the pita breads to form deep pockets. Spread 1½ tablespoons of sauce inside each pita. Layer in the steak slices, a spoonful of caramelized onions, and a handful of crispy potato matchsticks. Drizzle with more sauce, then repeat with another layer of steak, onions, and fries until full.

Serve immediately while warm. This steak pita is crisp, savory, and packed with flavor—ideal for a hearty lunch or satisfying dinner.