Preheat the oven to 400˚ F. Toss the pita pieces with 1 tablespoon olive oil and a pinch of salt on a baking sheet. Bake until golden brown and crisp, 5 to 7 minutes.
Meanwhile, combine the feta, garlic, lemon juice, 2 teaspoons za’atar and 3 tablespoons olive oil in a food processor. Puree until thick and smooth.
Whisk the pepperoncini brine, lemon zest and 3 tablespoons olive oil in a large bowl. Stir in the chopped pepperoncini and cherry tomatoes.
Place the steak pieces between 2 pieces of plastic wrap or parchment and pound with the flat side of a meat mallet or a heavy skillet until about 3/4 inch thick. Season all over with salt and pepper. Heat the remaining 1 tablespoon olive oil in a large skillet over medium-high heat. Add the steak and cook 3 to 4 minutes per side for medium rare. Transfer to a cutting board and let rest 5 minutes, then slice against the grain.
Add the romaine and toasted pita to the tomato mixture and toss; season with salt and pepper. Divide among shallow bowls. Top with the steak and sprinkle with more za’atar. Serve with the whipped feta.