Using a sharp knife, slice the tops and bottoms off of each orange, so they can stand up on the cutting board. Carefully cut the skin and the pith from the oranges, cutting from top to bottom. (The oranges should have no more white on them.) Slice each orange into thin round slices crosswise. Set aside.
Place the onions and 3 tablespoons white Balsamic vinager in a small bowl and let sit at least 15 minutes (up to 3 hours); remove the onion and discard vinegar.
In a medium-size bowl, whisk together the remaining 1 tablespoon vinegar with the olive oil. Toss with the arugla; season with salt and pepper.
Arrange the orange slices in alternating colors on a large plate. Sprinkle with mint, pistachios, and macerated onions. Pile the arugula in the center of the oranges and sprinkle with ricotta salata. Serve immediately.