Preheat oven to 450 degrees F.
Heat a cast iron skillet or grill to medium-high heat. Generously season the steak on both sides with salt and pepper, then rub all over with the 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.
Combine the garlic, 2 kinds of cheese, olive oil, lemon juice, red pepper flakes in a small bowl and whisk to combine. Taste and season with salt and pepper.
Divide the dough into 2 equal sized pieces. On a lightly floured surface roll the dough thin. Transfer to an oiled baking sheet, drizzle with olive oil. Bake 8-12 mins till golden. Repeat with the remaining dough and remove flatbreads to a flat clean surface.
Drizzle the flatbreads with a few tablespoons of parmesan pecorino dressing (reserving some to drizzle on top) Top with arugula and sliced red onions. Season with salt and pepper and drizzle with another few tablespoons dressing. Thinly slice the steak against the grain and arrange over dressed arugula. Drizzle with any remaining dressing, season with more salt and pepper and serve.