Preheat a dry large skillet, preferably cast iron, over medium-high heat.

Lightly score flank steak on both sides in a crosshatch pattern with a sharp knife.

Season steak generously with kosher salt and freshly ground pepper.

Rub the seasoned steak all over with vegetable oil.

Cook steak, turning once or twice to brown evenly, until medium-rare and lightly charred (8-10 minutes).

Transfer steak to a cutting board and let rest for at least 15 minutes. Reserve skillet.

Place thinly sliced red onion in a small bowl; cover with cold water and set aside.

Blend whole grain mustard and lemon juice in a blender to combine.

With the motor running, slowly drizzle in 1/2 cup olive oil until emulsified.

Gradually add finely grated Parmesan cheese and continue blending.

Stream in 1 Tbsp. cold water and blend until the dressing is smooth and thick. Season with salt and pepper.

Heat reserved skillet over medium heat.

Drizzle cut sides of ciabatta bread with olive oil.

Working with 1 half at a time, cook the ciabatta, cut side down, until golden brown and crisp (about 5 minutes each).

Place bread, cut side up, on a platter and drizzle with one-third of the Parmesan dressing.

Top with mature arugula leaves.

Drain the reserved red onion and scatter over the arugula.

Season with salt and pepper and drizzle with half of the remaining Parmesan dressing.

Thinly slice the rested flank steak and arrange over the dressed arugula.

Drizzle with the remaining Parmesan dressing and season with more salt.

Cut the assembled open-face sandwiches crosswise into pieces and serve immediately.