Preheat a grill, grill pan or large cast-iron skillet over medium-high heat. Brush the steak with olive oil and season with salt and pepper. Cook the steak, flipping, until medium-rare, 8 to 10 minutes. Transfer the steak to a carving board and let rest for 5 minutes.
Meanwhile, in a small bowl, whisk the sour cream with the horseradish and lemon juice. Season the dressing generously with salt and pepper.
Brush the flatbreads with olive oil. Working in batches, if necessary, cook over high heat, flipping, until lightly charred and crisp, 4 to 5 minutes. Transfer to a baking sheet or wire rack.
Thinly slice the steak against the grain. Spread some of the horseradish sauce on each flatbread and top with the steak, arugula, tomatoes and red onion. Drizzle with olive oil and sprinkle with salt and pepper. Cut the flatbreads into wedges and serve, passing the remaining dressing at the table.