Preheat the oven to 550°F or as high as it will go (if using a baking stone/steel; place the stone/steel in the oven before you start preheating).
In a medium bowl, combine the sliced steak with 2 tablespoons of the steak sauce. Toss well to coat. Set aside to marinate while you prep the other toppings.
Melt 2 tablespoons of the butter in a medium skillet over medium-high heat. Add the onions, a pinch or two of salt, and a few grinds of black pepper. Cook, stirring frequently, until the onions and soft and beginning to caramelize. Turn off the heat and set the onions aside.
Stretch or roll out your dough to a 12-inch circle, then transfer it to an oiled baking sheet or a floured pizza peel (if using a baking stone/steel).
Place the remaining tablespoon of butter in a small ramekin. Melt it in the microwave (about 20 seconds), then stir in the remaining tablespoon of steak sauce.
Brush the butter–steak sauce mixture on the dough, making sure to get all the way to the edges. Season with salt and pepper, then top with the onions and blue cheese.
Transfer the pizza to the oven and bake just until the cheese is bubbling and the crust has just started to turn golden, about 6 minutes on the baking sheet, 3 or 4 minutes on the baking stone/steel.
Take the pizza out of the oven. Arrange the steak on top, keeping the slices in a single layer. Return the pizza to the oven and bake for another 2 to 3 minutes or until the steak slices are brown on the edges but still a little pink in the center.
Take the pizza out of the oven. Sprinkle with the chopped arugula. Slice and serve, passing more steak sauce at the table.